Topologr. Vol. 25, No. 2. pp. I19- 153. 1986

Pmmd m Grcat Bnum.

@34-9383 86 $3.00 + W

F 1986 Perqmoo Press Ltd.

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES

c. T. c. WALL

(Receiced 4 February 1985)

ALTHOUGH the techniques of high-dimensional manifold topology have been successfully

extended by Freedman [ 143 to the topology of 4-manifolds, the results of Donaldson [ 1 l]

show that one must seek quite a different pattern in studying smooth 4-manifolds, for which

low-dimensional techniques may be more appropriate. Since our most coherent account of

three-dimensional topology is given by Thurston’s geometrization theorem [47], this

motivates the study of geometrical structures (in the sense of Thurston) in dimension 4.

By “geometry in the sense of Thurston” I understand a pair (X, G,) with X a l-connected

manifold, G, a Lie group acting transitively on X, such that:

(1) The stabilizer subgroup K, in G, is a point in X is compact (equivalently, X has a

G,-invariant Riemannian metric).

(2) G, has discrete subgroups r such that r\X (or equivalently, T\G,) has finite volume: i.e.

I- is a lattice in the sense of [40].

A manifold M has a geometric structure of type (X, G,) (or “modelled on X”) if M has an

atlas of charts mapping to X, with coordinate changes defined by elements of G,. Such a

structure is complete if it defines a homeomorphism of the universal cover fi with X, so that

M g r\X for r, a torsion-free discrete subgroup of G,. Equivalently [42, p. 4031 one may

talk in terms of a locally homogeneous metric on M, and metric completeness. In this paper

(except the final $11) we shall suppose M to be compact, so that completeness is automatic.

As the four-dimensional geometries have been recently classified by Filipkiewicz [ 133, we

embark here on the next step: the understanding of geometrical structures on closed

manifolds. One significant difference from the three-dimensional case emerges at once-in

most casts, a geometric structure carries a preferred complex structure, so we have a complex

surface. We are then able to appeal to the extensive existing classification theory available for

such surfaces, due for the most part to Enriques [12] (in the algebraic case) and to Kodaira

[29] (in general).

Indeed, the notion of geometric structures in this situation is no novelty. As Riemann’s

uniformization theorem showed that the universal cover of any compact complex analytic

l-manifold was biholomorphic to the projective line P i, the affine line 43 or the upper half-

plane H, work in this area is commonly known as uniformization theory, and there is already

an extensive literature. It is our contention that the approach here using Thurston’s ideas gives

a better model for understanding geometric behaviour than those employed earlier: for

example, affine structures (see $8 for references). As a first instance, observe that the groups of

complex-analytic automorphisms of Pi, C and H form, in each case, Lie groups whereas this is

no longer the case for C’, for example [consider the automorphisms (x, J) -, (x, y + 4(.x)), 4

being any entire function]. For a compact surface with a complex analytic affine connection, it

does not even follow that the universal cover is the whole of C*. These remarks suggest that a

geometric structure in the sense defined below may impose so stringent a condition that it

TOP 25:*-a 119

120 C. T. C. Wall

excludes large classes of surfaces. However, while this is certainly true, these same classes are

excluded by many of the older theories: the rigidity of Thurston geometries is largely

compensated by their variety. Moreover, we will show that the assignment of the appropriate

geometry (when available) gives a detailed insight into the intrinsic structure of the complex

surface.

The plan of the paper is as follows. The first three sections are devoted to an abstract study

of the geometries in question: in $1 we give the list, and also the list of compatible complex

structures as determined in our previous paper [49]: here we give explicit models. In $2 we

apply general results about lattices to the cases in question; these results are used in 93 to give

a more accurate listing of the geometries themselves (the original list in $1 gives only the

maximal connected group G ,T).

Next, $4 recalls the main features of the Enriques-Kodaira classification in a convenient

form. This classification is not as well known as it should be, and the results, beyond Kodaira’s

original papers [ZS], [29], are rather scattered, so we have presented (partly in $4 and partly

in the next five sections) a rapid survey of the results. The recently published book [23 is also

valuable in this area.

In $4 also we state our first main theorem: that a geometric complex surface has its place in

the classification determined by the geometry X. Then in the main body 95-59 of the paper,

where different cases are discussed in turn, we prove this theorem and also discuss in each case

necessary and sufficient conditions for a compact complex analytic surface to carry a

compatible geometric structure: our best results are obtained for elliptic surfaces. These results

are mostly already known in some form or another.

In $10 we extend our conclusion by contemplating geometric structures that need not be

compatible with the complex structure, and also show that if a closed manifold M4 admits a

geometric structure of type X, then X is already determined by the homotopy type of M.

Finally in $11 we collect miscellaneous observations about possible further lines of

investigation.

$1. DESCRIPTION OF THE GEOIMETRIES

Throughout this paper, the terms “geometry” and “geometric structure” will be used in the

sense of Thurston: see Scott [42]. To rephrase slightly the above definition, we have a pair

(X, G) where:

(i) X is a complete, simply-connected Riemannian manifold;

(ii) G is a group of isometries of X;

(iii) G acts transitively on X; and

(iv) G contains a discrete subgroup I- with r\X of finite volume.

The metric is of use in describing the conditions above, but is not regarded as part of the

geometric structure: note, however, that it implies that G acts with compact stabilizers. Thus

we regard (X, G) and (X’. G’) as defining the same geometry if there exist a diffeomorphism of

X on X’ and an isomorphism of G on G’ taking the first action to the second. Observe also that

if G is contained in a larger group G” of isometries of X, the geometries (X, G) and (X, G”)

are very closely related and we shall often identify them. These situations will be further

explored in $3 below: until then we suppose (except when otherwise specified) that

G = GO, is the maximal connected group in its equivalance class acting on X.

We now describe the geometries in dimensions 5 4. In dimension 2, these are well known;

the classification in dimension 3 is due to Thurston (see Scott [42]) and in dimension 4 to

Filipkiewicz [I33 (I believe that there is also an account in lecture notes of Kulkarni).

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 121

Dimension 1. There is a unique geometry: that of the (real) Euclidean line: we denote it by

E’.

Dimension 2. There are three geometries: those of the sphere S’, the Euclidean plane E2

and the hyperbolic plane H2.

Dimension 3. Again we have the sphere S3, Euclidean space E’ and hyperbolic spaz H3.

Next we have the products S’ x E’ and H2 x E’. The universal (infinite cyclic) cover SL, of

the unit tangent bundle PSL2( R) of H’ is a group, and left translations give isometries, as do

right translations by the induced cover (isomorphic to W) of PS02: thus the isometry group

can be written ST2 x ,W.

We also have the group R;i13-the unique simply-connected nilpotent group of this

dimension-this has a circle group of outer automorphisms which yield isometries. Finally

there is a solvable group SO/~, the split extension W2wZW where the quotient W acts on the

subgroup R2 by

a(t) (x, y) = (e’x, e-‘y).

Dimension 4. Here we have the irreducible Riemannian symmetric spaces S4, H”,

P2(C), H’(C) = SU2,1/S(CJ2 x U,), and a list of products of lower-dimensional geometries:

S2 x S2, S2 x E2, S’ x HZ, E“, E’ x H2, H2 x H2, S3 x E’, H3 x E’, ST, x E’, Nil3 x E’

and SO/~ x EL. There are the nilpotent Lie group Nil4 = R3xp W and the solvable Lie groups

sol;. * = w3=7_,” R where the action of the quotient is given respectively by

B(rl=exp{c( a g X)}, im,“(r)=exp{c( I i 8)}

where a > 6 > c are real, a + b + c = 0, and en, e*, ec are the roots of i.3 - mi.* + ni. - 1 = 0

with m and n positive integers. If m = II, then b = 0 and we can identify Sol;‘,. m = SoI3 x E’ : in

general, Sol’,, n and Sol$. n, are isomorphic iff the corresponding matrices are proportional.

The case of equal roots is excluded here, but the semidirect product Sol: = W3~6 R with

6(t)(x, J, z) = (e’x, e’y, ee2’z)

gives the next geometry: observe that there is a further circle of isometries here rotating the

first two coordinates. There is a further solvable group Sol;, conveniently represented as a

matrix group

ct, a. b, c E 19,

Finally we have the geometry F” with isometry group [R’x SL2(R) [with the natural

action of SL2( W) on R2] and stabilizer S02(R). This is the only geometry in the list to admit

no compact models.

The list of geometries arranged by isotropy subgroups is shown in Table 1.

In the previous paper [49], after describing this list, I proved the following:

THEOREM 1.1.

(a) X carries a complex structure compatible with GO, if and only if it is one of P2(C), H’(C),

S’ x S’. S2 x E2, S2 x H2, E2 x H’, H’ x HZ, F3. ST2 x E’, Nil’ x E’, Sol%, Sol:. I

122 C. T. C. Wall

Table 1

(b)

(4

(4

Isotropy Geometries

S’, E4, H4 P=C. H=C S= x S2. S2 x E=, S= x HZ, EL x HZ, Hz x HL Szx E’, H’ x E’ SL, x E’, Nil’ x E’, So14, F4 NiP, Sol:. I (including Sol’ x E’). Sol:

Here (S’),,,. denotes the image of S’ in (i, c SO1 by z ~(z”‘, z”).

The geometry E4 (resp. S3 x E’) carries a complex structure compatible

group R4ixU2 (resp. U2 x W), which still defines a geometry. with the smaller

The remaining geometries S4, H4, H3 x E’, Nil4 and Sol:, n do not admit a complex structure compatible with the geometric structure.

In each case except Sol;, the complex structure on the maximal relevant geometr? is unique

up to isomorphism. For Sol;’ rhere are just two isomorphism classes.

This result was proved in [49] using Lie algebra calculations. The statements about

existence or non-existence will be given alternative (less direct) proofs below. The same

comment applies to the next result.

THEOREM 1.2. The geometries in the list

P’(C), H’(C), S2 x S2, S2 x E2, St x H2, E4, E2 x HZ, HZ x HZ, F4

admir a Ktihler structure compatible with the maximal relevant group of isomerries. In the

remaining cases of Theorem 1.1 (b), (c)

S3 x E’, Nil3 x E’, S^vL2 x E’, Sol:, Sol:, Sol’:,

there is no Kiihler structure comparible with a geometric structure.

Rather than repeat the earlier proofs, we now exhibit explicitly the complex and Kghler

structures in question in the cases when they do exist.

We begin in fact by noting that the two-dimensional geometries S’, E2 and H’ can be

naturally identified with the complex projective line P’(C), afline line b: and upper half-space

H’(C) (denoted hereafter by H) and so acquire (complete) invariant Kghler metrics: hence so

do their products. The geometries P2 (C) and Hz(C) are well known to be KIhlerian symmetric

spaces (and we shall not need the formulae here).

For the case of F4, first define an action of Rz~SL2 on [w2 x Hz as follows. Take (x, y) as

coordinates in W2, and z (with Im: > 0) as (complex) coordinate in H. Then define an action

of R2 by

(u, v)(x, y, 4 = (u + x, v + y, z)

and an action of SL2 (W) by

(: ~)(x,?.r)=(ax+bg,cx+dy.~).

It is at once verified that these combine to give an action of Iw2=SL2 such that the stabilizer of

(0, 0, i) is SO2 c SL2. The action does not preserve the complex structure given by taking

GEOMETRIC STRUCTURES OS COMPACT COMPLEX ANALYTIC SURFACES 123

x + iy and z as complex coordinates: instead we set w = x - JZ. Then the action becomes

(U, c)(w, z) = (U - t’z + w, -_)

(; ;)(w.z)=(-$~)

and so does preserve the complex structure defined by taking w E I= and z E H as coordinates. It

is not so easy to write down a Klhler structure, but following through the construction given

in [49] leads to the formula

(Imz)-2(dz@d~)+(Im:)-’ Imw

dw-=dz ).i

@

S’ x E’. We identify (using polar coordinates)

S3 x E’ = R4 - (0) cz C2 - {0)

and give this the obvious complex structure. It is invariant under U2 x R, which UZ acts

linearly on C2 and R by homotheties.

Nil3 x E’. Define a multiplication on C2 by

(w, z)(w’. 2’) = (w + w’ - iZz’, z + z’).

It is easy to verify that this satisfies the group axioms. The commutator of (w, z) and (w’, z’) is

(i(iz - Tz’), 0); this lies in the one-dimensional central subgroup W x 0: the central subgroup

ilw x 0 splits off as a direct factor. The group is thus isomorphic to Nil3 x E’. The group SO2

of outer automorphisms acts by

t(w, z) = (w, tz) ([EC, ItI = 1)

so this also preserves the complex structure on C*.

ST2 x E’. We have natural actions of SL2(R) on H and on its bundle of non-zero

tangent vectors: this lifts to the universal covers. Indeed, the action

yields, on differentiation,

azfb = = -

cz + d

( ) ; f; (3 = a- To obtain the universal cover, write w to represent log (dz): then we have

(1 ;)(w, -_)= (w-Uog(cz+d),=$).

The isometry group of ST2 was Sy2 x ,R. The central subgroup Z here corresponds to adding

2x to the value of the logarithm: the group W acts by translating the w coordinate by an

imaginary quantity. Translations of the other factor E’ will translate w by a real amount.

SO/~,. Here we have a normal abelian subgroup of rank 3, which we identify with C x W,

and a quotient group SO2 x R, which we identify with 6: ‘. Define actions on C* by letting

+ Properties of this geometry are also discussed in R. BERNDT: Some differential operators in the theory of Jacobi forms. IHES, preprint, (March 1984). In particular, this metric appears there on p. 8.

124 C. T. C. Wall

(a,b)E6Zx!Zandi.~2=“acton(w,c)E~xHby

(a, b)(w, z) = (w + a, : + b)

j.(w, 2) = (E.w, lLI-2z).

This gives an action of the desired group which is transitive, preserves the complex structure,

and has trivial stabilizer.

Sol:. This group was already presented as a matrix group: again take (w, Z) as coordinates

in Q) x H, and define an action by

(; ; y)!,) =(“:1;;‘).

Again this is transitive and has trivial stabilizer. The other complex structure, which we denote

by SoYi4, is obtained by modifying this to give

(w+bz+c+iloga,az+a)

(note that a > 0: logr denotes the real logarithm). The verification of compatibility is again

trivial.

The above explicit models give us the global complex structures of the various model

spaces X.

For the case with real dimension 2, we have already identified

S2 = P’(C), E2 = C, H2 = H’(C) = H.

Products of pairs of these are self-explanatory, as is P2(C); H2(C) has a standard realization as

the unit ball Izl’-t 1~1’ < 1 in C2. The rest were identified above (up to holomorphic

equivalence) as

S3 x E’ z C’ - {0}

Nil3 x El z C2

ST, x E’ g So14, z Sol: z Soli Z F4 2 C x H.

In particular, we have

COROLLARY 1.3. The ten complex two-dimensional geometries other than P2(C), P’(C)

x P’(C), P’(C) x C and P’(C) x H are biholomorphic to domains in C2.

$2. PLACEMENT OF DISCRETE SLJIBGROUPS

We collect here some general results about lattices in connected Lie groups which give

some geometrical insight into the corresponding quotients, and which will be of use in the

sequel.

Our general reference is Raghunathan [40].

Recall that we have a l-connected manifold X, and a connected Lie group, here denoted

G,, acting transitively on X such that the isotropy subgroup K, is compact. Thus if F is a

discrete subgroup of G, F \X is of finite volume (resp. compact) if and only if F \G,is SO: thus

I is a lattice in G, in the sense of [40]. Now we have:

2.1. [40, 3.11. IfG is solvable, then any lattice r in G is cocompact.

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 12 j

2.2. (Mostow) [40. 3.31. IfG is solcable, wirh nil radical N. and r is a lattice in G, rhen

r n ,V is a lattice in S.

2.3. If G is nilpotent, brith derived subgroup G’, and I’ is a lartice in G, then r n G’ is a latrice

in G’.

Proof. We can consider G as an algebraic group (cf. [40, 2.21 j---then F is Zariski-dense

C40.2.31. It follows that [I, I] is Zariski-dense in [G, G] = G’and hence [40. 2.31 cocompact

in G’. Hence so is F n G’, which is of course still discrete.

2.4. (Borel). If G is semisimple and algebraic, and r a lattice in G, then I- is Zariski-dense in

G. Any algebraic subgroup normalized by I- is normalized by G. If, moreover, G has no compact

connected normal subgroup, the centralizer of I- is the cenrre of G.

Proof. [-IO, 5.4, 5.16, 5.181.

2.5. (Bieberbach). If r is a discrete subgroup of the isometry group of E”, with compact

quotient, then r contains a lattice of translations of rank n (and finite index in r).

Proof. [40, 8.261.

2.6. (Wang). If G is a connected Lie group wirh radical R, such that G,‘R has no compact

factor, and lI a larrice in G, then I- n R is a larrice in R.

Proof. [40, 8.271.

Most of the above results give us a normal subgroup H of G such that F n H is a lattice in

H. It follows (indeed, is equivalent) that the image of F is a lattice in G/H. When G acts on X as

above, we have a fibration X + X/H, and the fact that HT is closed shows that we have an

induced Seifert fibration

I-\X( = I-\G/K) + l-\X/H( = T\G/HK).

We apply these to all the geometries of dimension I 4.

Euclidean cases

For E2, E3 and E’, (2.5) applies: F has a translation subgroup of finite index.

Nilporenr cases

For Ni13, Nil3 x E’, Nils, (2.3) applies. Here, F n G’ is a group of translations

(of rank 1, 1.2, respectively; the quotient is a subgroup of a vector space, so again a lattice of

translations (of rank 2, 3, 2).

Solcable cases

For So13, Sol:. n, Sol: and Sol?, (2.2) applies. In each but the final case, the nilradical is

Euclidean: thus F n N is a group of translations (of rank 2, 3, 3) and the quotient is infinite

cyclic. If G = Sol:, then N z Ni13: thus F n N is a 1a:tice in Nil3 (as discussed above); the

quotient is a lattice in R x SOz.

Mixed cases

For Hz x E’, Sy2, Sy2 x E’, H3 x El and F4, (2.6)applies. (Note that we have IistedX and

not G,.) The radical is Euclidean, of respective dimensions 1, 1, 2, 1, 2; so F n R is a group of

126 C. T. C. Wall

translations of the corresponding rank. The quotient is a lattice in PSL,(R) in all of these

cases except H3 x E’.

An alternative, more geometrical proof of this conclusion for Hz x E’ and ST2 has been

given by Thurston ([47]; see also Scott [42, pp. 461, 4661). This method of proof was

extended in [49.6.3] to give corresponding results for ST1 x E’, Nil’ x E’ (as above) and also

HZ x E’.

Semisimple cases

For H2, H3. H’, Hz x H2 and H’(C), (2.4) applies: note, moreover, that in each of these

cases the group G has no compact factor. These are the cases giving the richest and most

interesting variety of geometric structures. This is in contrast to compact cases.

Compact cases

For St, S3, SJ, S2 x S2 and P2(C) the group I must be finite, and all cases may be

enumerated.

Remaining cases

These are S’ x E’, S2 x E’, S2 x H2, S3 x E’; in each, X is of the form A x B with A

compact. Thus G, and hence r act properly on B: I is obtained by lifting this action

(arbitrarily) and combining with a finite isometry group of A, normalized by this lifted group.

Only for the final group of geometries is X non-compact and not contractible: indeed,

in all the other non-compact cases, X is homeomorphic to Euclidean space of the same

dimension.

The grouping into cases given above gives a better insight into the structure of the several

geometries. Combining with the results of $1 yields Table 2.

Table 2. Summary list of geometries of dim < 4

Dim4

Dim2 Dim 3 KHhler Complex non-KHhler Non-complex

Compact S’ S’

Compact factor S’ x E’

Euclidean

Nilpotent

Solvable

E2 E’

Nil”

SOP

Mixed

Semisimple

H,’ x E’

SL*

H= H’

P’(C) s x .s S” x E’ SL x Hz

S'

S'x E'

E4

Nil’ x E’ Nil’

so14, sol:. I sol;

Hz x E2 ST, x E’ H’XEL F4

H*W H4 HZ x Hz

$3. MAXIMAL AND MINIMAL GEO.METRIES

In our listing of geometries in $1 we described, for each model X, the maximal connected

group GO, of isometries of X. In the first part of this section we will list those connected

subgroups H, of GO, such that (X, H,) is also a geometry: this result was in fact used in [49]

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 127

and we shall make further occasional reference below. In the second part we will determine

the full maximal group G, of isometries of X.

Such descriptions seem necessary for an adequate understanding of the geometry X. Our

results in the second part are relevant to the questions of orientation-reversing homeomorph-

isms and of anti-holomorphic equivalences: particularly complex conjugations giving real

forms. The non-maximal geometries of the first part, though they introduce an annoying

complication into the classification, are important in applications as the following four

paragraphs illustrate.

It is shown in [42. pp. 445,455] that a closed 3-manifold with geometric structure of type

S3 resp. E3 in fact admits one of type (S3, U,) resp. (E3, E3 =SO,): it is these latter which

correspond to Seifert fibre space structures. Similar examples in dimension 4 may be found

below.

We saw in $1 that two geometries (S3 x E’ and E4) only admit a compatible complex

structure when the group of the geometry is restricted (to U2 x R or E4 w U2, respectively).

A weighted homogeneous normal surface singularity admits, according to Neumann [38],

a natural geometric structure ofone of the types S3 x E’, Nil3 x E’ or SE2 x E’. It was shown

by Dolgachev [lo] that the singularity is Gorenstein if and only if there is a structure of the

corresponding type (X, H,), where H, 2 X acts on itself by translations.

We shall see in $9 below that manifolds with geometric structure of type So/i are Inoue

surfaces. In each case. the geometric structure can be taken to be non-maximal.

THEOREM 3.1. The non-maximal geometries with connected isometry group in dimension

I 4 are as follows:

Dimension 2: (E*, E’)

Dimension 3: (S3, U,). (S3, SU*), (E3, E3 WS02), (E3, E3), (ST,, S?,), (Ni13, Ni13).

Dimension 4: (E4, E*xK); K = U2, SU2, SOJ, Sgz x S02, S02, (S’),,,, (l).; (S’ x E2,

SO3 x E’), (H’ x E 2, PSL2 x E’), (SL, x E’,Sy, x E’), (S3 x E’, U: x E’),

(S3 x E’, SU2 x El), (Nil3 x El, Nil3 x El), (Sol:, Hz) (as listed below).

Proof. We organize the cases following the discussion in 92 of the different sorts of

geometries.

Euclidean cases

Since any r has a translation subgroup of finite index, H, must contain E”. It follows that

H, is a split extension of E” by a maximal compact subgroup K (the splitting follows since

such a K has a fixed point on X): K can be any (connected) subgroup of SO,.

Nilpotent cases

As Nil’ already has trivial stabilizer, there can be no non-maximal geometry here. In the

cases of Nil3 and Nil3 x E’ the stabilizer of GO, is S02, hence that of H, must be trivial. Now

by (2.2)any lattice Tin GO, meets Nil3 resp. Nil’ x E’ in a lattice. Thus H, must contain. hence coincide with Nil3 resp. Nil3 x E’.

Solruble cases

For So13. Sol:. n and Sol’: the connected isotropy group is trivial and there is nothing to

prove. For So/z we know by (2.2) that any lattice I meets R3 in a lattice in R3: thus H,y must

128 C. T. C. Wall

contain W3. Since the stabilizer of GpY is a circle, that of H, must be trivial. Thus H,Y is an

extension of R3 by a l-parameter subgroup L of the quotient W x SO2 z C ‘. Conversely, for

such an extension to be a geometry, there must be a lattice in R’ invariant under some element

i. E C ‘. Now the eigenvalues of i. on R3 are i., /trand 1 i. I- 2: for there to exist an element of

SL,(Z) with such eigenvalues, j. must be a unit in a complex cubic field.

Semisimple cases

If GO, is semisimple with no compact factor then by (2.4) if T is a lattice in H, c G: then as

T normalizes H,, so does GO,. Thus H, is normal and Zariski-dense in GO,, SO H,y = G,.

Mixed cases

If GO, is an extension of a vector group V by a semisimple group G with no compact factor,

and r a lattice in G$, then by (2.6) the image of T in G is a lattice in G, so arguing as above, A

= G. Hence H, is an extension of H, n V by a Levi complement for H. In the cases X = H’ x E’, H3 x E’, F4 this is isomorphic to fi. But H,n I/ contains the full group T n V of

traniations hence (being connected) is the whole of V. Thus again H, = G,. In the case X

= SL,, the subgroupS?, covEing Galready contains a lattice in V, so defines a non-maximal

geometry. Similarly for X = SL, x E’ we have non-maximal geometries with H, = SL, x W

for some line W c V. All these cases are isomorphic.

For the remaining case X = H2 x E2 in this group, we know that r n E2 gives a lattice in

E2 and that the projection of r on PSLz defines a lattice there (see [49, 6.31). As above, it

follows that H, must contain PSL2 x E2, and so equals this if it is not maximal.

Compact factor cases

If X = S2 x B, then the projection of H, on Isom (S’) must be transitive, hence the whole

of S03. As SO3 is simple, and does not occur in Isom (B), it follows that H, = SO3 x C, where

(B, C) is a geometry. A similar argument works for the S’ x E’ case to show that our geometry

is a product of non-maximal geometries.

Compact cases

The only condition on H, here is that it acts transitively on X. If X = S’, then clearly H, must be S03. For X = S3, Hx/{+ 1) c SO3 x SO, must project onto at least one of the

factors: this leads quickly to the conclusion. Again for X = S2 x S2, SO3 x SO3 has no

transitive subgroup. In the remaining cases X = S4, P2(C), G, has rank 2: here a subgroup of

rank 1 is not large enough to be transitive, nor is one of rank 2 (necessarily S’ x S’, S3 x S’ or

similar).

Collecting the results obtained above yields the result summarized in Theorem 3.1. The

results were summarized rather inaccurately in [49]: in particular, the case (i\*i? x E’)

mentioned there does not admit a lattice. It is worth noting that apart from the somewhat

exceptional case of Sol:, the non-maximal geometries all belong to the family of Seifert fibre

geometries.

We turn to the listing of maximal geometries. Here we describe the methods first and

summarize the results at the end.

First, if we denote by KO, the isotropy group of GO,, then a group G, with identity

component G “x will have stabilizer K, c 0, with identity component KO,, and hence

normalizing KO,. This already enables us to deal with a dozen cases.

If K i = SO, (cases X = S4, E4 or H4) then K x = 0, as in each case X admits a reflexion.

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 129

IfKO, = Lj2 [cases X = P’(C), H’(C)] then K,Yequal~ the normalizer, c2,say, of C2 since

in each case X admits a complex conjugation.

If KO, = SO2 x SO2 then K,x = O2 x O2 (cases X = S2 x E’, S2 x H2, E’ x H2) or, if there

is an isometry interchanging the factors (cases X = S2 x S’, H’ x H’), K, = H2qti02. If KO, = SO, (cases X = S3 x E’, H3 x El) then K, = O3 x Oi: again the existence of the

desired reflexions is clear in each case.

In general, if we have a Riemannian product A x B where A and B are irreducible and not

isometric to each other, then, since such (de Rham) decompositions are unique we have (see,

for example, [27. Vol. 1. p. 2401 Isom (A x B) = Isom A x Isom 3. This line of argument

recovers some of the cases above and also deals with Nil3 x E’, where K, = 02 x 0,; with

ST2 x E’, where again K, 2 O2 x 0,; and with Sol3 x E’, where K, z D8 x Oi (here DB

denotes the dihedral group of order 8): beware that these isomorphisms do not make explicit

the representation K, -. 0,. In the remaining cases, we calculate directly:

F’. Any automorphism r of GO, = W2~.SL2 (W) must leave R2 invariant. As complements

to W2 in the Lie algebra are all conjugate by R2, we may suppose cx also leaves SL2(IR)

invariant, Now rl W2 defines an element of GL2(W), whose class modulo .SL2(?2) is unaltered

on changing r by an inner automorphism. If this class is trivial, we may suppose r the identity

on iw’: but now rlSL2( 2) is also the identity, since an element of SL,(W) is determined

by the way it acts on II@. In general, as we are only interested in r of finite order, det (xl W2)

has finite order, thus equals f 1. There are thus essentially only two cases for x. and

G, = Wz= SL2 ‘(W), where the + denotes that matrices may have determinant f 1.

In the remaining cases we calculate in the Lie algebra. Since we are seeking a compact

automorphism group, which will act semisimply, any invariant subspace will have an

invariant complement and we suppose our bases chosen to fit these: this explains the

simplifications employed below.

Sol:. GO, is an extension of W3 by R x S02. Since W x SO2 acts faithfully, any

automorphism x not only preserves W3 but also is determined by its effect on R’. Further, cz

preserves the decomposition as W2 x W into irreducible SO,-modules. Thus xl W3 E O2 x Oi,

and since these automorphisms are realizable, we deduce K, g O2 x Oi.

The last three cases all have K “x trivial.

Ni14. The Lie algebra g has a basis {ei, e2, e3, e4j with Lie bracket defined by [e,. ei]

e2, [et, e2] = e3, other [ei, ej] = 0. Thus r preserves g’ = ( e2, e3 ), Z(g) = ( e3 );

c:mplements to these (e,, e4) and (e,, e2, e4) so reduces to

cf(ei) = aei +be, r(e2) = re2

z(e4) = ce, + de, r(e3) = se3.

This defines an automorphism only if r = ad - bc # 0, s = dr # 0. In partitular, d # 0:

we may thus suppose b = c = 0 and then take a, d = + 1.

Sol:,,. Here g has basis {e,) with

Ce4, ell = w, Ced, e21 = bez, Ce4, e31 = c-3.

Again r preserves (e,, e2, e3 ), and indeed each of the eigenspaces (ei ) of ad e4. We may

suppose %(e4) = e4, and then r(e,) = _+e, for the rest. The case Sol’ x E’ is different here

since the eigenvalues of - e4 are (in this case only) a permutation of those of e,. so we have an

extra automorphism.

Sol;‘. The Lie algebra g is defined by

Cei, e21 = e2, Ce,, e31 = -e3, [e2. e3] = e4.

130 C. T. C. Wall

Thus g’ = (e2, e3, e,), g” = Z(g) = (e,): our automorphism x fixes these and their

complements, hence (e, ) and ( e2, e3 ), and indeed preserves or interchanges the

eigenspaces (e,. ), ( e3 ) o a f d e, on the latter. If r preserves all coordinate axes, we find %(.s, r~)

with

r(el) = el, de21 = -2, r(e) = w3, rl(e4) = tqe4 (E, 9 = + 1);

we also have the interchange

r(ei) = -el, r(e2) = e3, r(eJ) = ez, r(e4) = -e4.

Thus KO, 2 Ds.

We now summarize these results.

THEOREM 3.2. (a) The stabilizers K, of the maximal geometries are gicen (up to

isomorphism) by Table 3:

Table 3

Kx X Kx X

0. S4, E'. H4 02x0, Nil3 x E’, ST, x EL, Sold 02 f”(C), H’(C) 02 F4 Gl.pz S= x S’, Hz x H= De x Ez Sol3 x E’

02 x 02 S2 x E’, S2 x HI. EL x H' D8 sol: 03x01 S' x E’, H' x EL 22 x z2 x 22 sol:, m

Zl x hz Nil4

(b) There exists an orientation-reversing element of K x in all cases except P2 (C). H’(C) and

F4.

Conclusion (b) is obtained by checking the above descriptions in each case: for example, with

Sol;’ the automorphisms cz preserve orientation: r changes it.

For a second summary we consider the effect on our complex structures. First, we

complete the above by determining the maximal K, for the relevant geometries (E’, El= UI)

and (S3 x E’, U2 x W). In the first case, K, = 0, [argument as for P’(C)]. For the second, it

suffices to calculate the normalizer of U2 x R in 0, x Isom W: this equals 0, x Isom R, with

just four components.

THEOREM 3.3. (a) For the 15 complex geometries (X, GO,) of Theorem 1.1, the group

CC of holomorphic automorphisms has two components if X = P’(C) x P’(C), H x H, or Sol:,

and only one connected component in the remaining cases. (b) In all cases there exists an anti-

holomorphic automorphism.

Proof. For P’(C), C2 and H’(C) the unique non-identity component (of o,, for example)

is anti-holomorphic. For the five other products of two two-dimensional geometries, the

interchange of factors (when possible) is holomorphic; conjugation in both factors together is

anti-holomorphic: the other components reverse orientation.

For the three Neumann geometries S3 x E’, Nil3 x El and Sz, x E’, K, has four

components. Of these, two reverse orientation: the other non-trivial one is anti-holomorphic.

This is immediate for S3 x E’; for the others it follows easily from the Lie algebra calculations

of [49]. Indeed, for Nil3 we had El = e, + ie2, E2 = e3 + ie,; and we have the automorphism

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 13 1

changing the signs of e, and e3; similarly for STl x E’ we have an automorphism replacing

each of El, El as given by their complex conjugates.

In the cases of F’, the unique non-trivial component is anti-holomorphic. Perhaps the

easiest way to see this is to note directly that F4 admits the conjugation (w, :) + (H.. - 3,

and that this induces the automorphism of GO, under which (u, C) + (u, -c) and

(z :)-( -: -i).

There remain the cases Sol;‘,, Sol: and SO~\~. For So/i, the group of components is

generated by the elements r and B which reverse the signs of ez and e3, respectively.

Thus the non-trivial orientation-preserving case is r/?, which indeed takes both El = el + ie,,

E, = e, + ie, to their complex conjugates. For Sol;, T changes orientation; a(1. - 1)

is holomorphic and z( - 1, 1) anti-holomorphic: recall that here E, = e, +ie, and

E2 = e3 + ie,. In the case of So/i4, where El is redefined as e, + i(ez + e4), z( - 1, 1) is still

anti-holomorphic, but ~(1, - 1) is no longer holomorphic. This concludes the proof ofTheorem 3.3. We remark that in all except the final case. every

orientation-preserving element of G, is either holomorphic or anti-holomorphic.

94. CLASSIFICATION OF COMPACT COMPLEX ANALkTIC SURFACES

As it is our objective to relate geometric structures to the Enriques-Kodaira classification

([12], [29]), we now recall the main features of the latter. The most convenient presentation

of this uses several more recent results also: I endeavour to give precise references, since these

are not always easy to find, though the recently published [Z] is very useful: see also [48].

There are two independent principles of classification, which we discuss in turn,

corresponding to two invariants: the Kodaira dimension x and the first Betti number bl

modulo 2.

Let V be any compact complex manifold, K its canonical bundle (of holomorphic n-forms,

where n = dim V). The plurigenus P,,,(V) is the dimension of the space of holomorphic

sections of the mth tensor power mK of K. The Kodaira dimension K(V) is defined [22] by:

if P, = 0 for all m 2 1, then K(V) = - 1 (many authors write - 2) otherwise, K(V) = lim SUP~_~ (log P,,,/logm).

The following equivalent forms of this definition can be extracted from [22] (though they are

not explicitly stated there):

ti( V) = sup,{dim$,( V): c$~ the map to projective space induced by mK}.

1 + X(V) = transcendence degree over C of the “canonical” ring @,,, t oH”( V; mK).

Thus the only possible values of K( V) are the integers r with - 1 I r I n. It also follows from

[22, Theorems 1 and 23 that:

given V with .v( V) 2: 0, there exists a positive integer m, such that Pk,,J V)/(km,) K(” tends

to a positive non-zero limit as k + cc through positive integers.

Iitaka also shows (lot. cir.) that ti( V) is a birational invariant, and proves

PROPOSITION 4.1. [22, Theorem 61. If ? is a finite unramijied covering of V, then

h.(F)= K(V).

For the special case of surfaces there is a further invariance property.

PROPOSTION 4.2. [21, II]. For S a compact complex surface, the plurigenera P,,,(S). and

hence x(S) are invariant under deformations of S.

132 C. T. C. Wall

The Betti numbers b,(S) of a compact complex surface S are defined in the usual

topological sense (with, say, real coefficients). The importance of b 1 (S) mod 2 was recognized

by Kodaira. Clearly, it is invariant under deformations! More significantly, we have

THEOREM 4.3. The following conditions on S are equicalent:

(i) bl(S) is ecen.

(ii) S deforms to a projectice algebraic surface.

(iii) S has a Ktihler metric.

Proof. For the equivalence of(i) and (ii), see Kodaira [29, Theorem 251: their equivalence

with (iii) was also conjectured by him. That (iii) implies (i) follows from elementary Hodge

theory.

For the converse, we refer to the classification in [29, Theorem 221. (We could use instead

Theorems 4 and 11). For class I, P9 = 0 so by [29, Theorem lo] these surfaces are all projective

algebraic, hence Klhler. Class II consists of K3 surfaces, which have Kahler metrics by Siu

[44] (see also [4)). Class III consists of complex tori, which are well known to possess such

metrics. Class IV consists of elliptic surfaces: the existence of a KChler metric here (assuming

bi even) was established by Miyaoka [32]. Finally Class V consists of surfaces with x = 2,

which are projective algebraic (by [S]: or see [29, Theorem 91).

The invariant b,(S) mod 2, also, is stable under finite coverings.

PROPOSITION 4.4. Ifs’ is a finite unramifed covering of S, then s has a Klihler metric if and

only if S does.

For the pullback to $of a Kahler metric on S gives a Kahler metric on s’. For the converse,

choose a covering Sof s’ which is a regular covering of S. Then a Kahler metric on s’ pulls back

to one on S. Take the average of the transforms of this by the group of covering

transformations of S over S. All these are Klhler; so is their average, since (i) positive

definiteness of the real part defines a convex cone; and (ii) closedness of the imaginary part

is a linear condition. We thus have an invariant Klhler metric on S: this projects down to give

a Kahler metric on S.

Since (as above) h’(S) = 2 implies bl (S) even, we have defined seven classes of surfaces.

These do nor coincide with the seven classes defined by Kodaira ([oc. cit.) which do not, for

example, enjoy the property of stability by finite unramified coverings. Various further

properties are known about these classes, which will be mentioned in the appropriate sections

below.

Our interest in these classes was motivated by their connection with geometric structures

in the sense defined above. Our main conclusions can be (in part) summarized as follows:

THEoREM4.5. Let S be a compact manifold with a geometric structure of type X. Then S has

a compatible complex structure with incariants related to X as in Table 4:

Table 4

b, even b, odd

K= -1 S’ x S’, S2 x E’, S2 x HZ, P’(C) S’ x E’, So/‘,, Sol; K=O E4 NiP x E’

h.=l E’ x H’ ST, x E’ x=2 H’ x H’, H’(C) -

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 133

COROLLARY 4.6. If X is one of S’ x E’, Nil3 x E’, ST2 x El. Sol4,and Sol:, then X does

not possess a K&ier structure compatible wirh the geomerrg.

For such a structure would be inherited by surfaces modelled on S. contradicting the

theorem. This proof is perhaps simpler, and certainly more enlightening, than the direct

computation in [49].

In addition to proving this theorem we will seek to characterize, within each class of

compact complex surfaces, those that possess a geometric structure: the results are precise in

most cases.

The various cases will be treated in sections below as indicated in Table 5:

Table 5

K

b, even

b, odd

-1 0 1 z

§S $8 $7 $6

99 47 97 -

The uniqueness of geometric structures will be discussed in $10.

$5. RULED SURFACES

The conditions K(S) = - 1, b,(S) even are known to characterize the class of ruled

surfaces: surfaces admitting a family of embedded copies of PI(C) whose union is dense.

Given a copy of Pi(C) in a surface S with geometric structure, having X as universal cover,

since P,(C) is simply-connected, the embedding of it in S lifts to one in X. Now since all global

holomorphic functions on Pi(C) are constants, there can be no embedding of X in C2.

By Corollary I.3 the only possibilities for X are now P2(C), P’(C) x P’(C), P’(C) x C and

P’(C) x H. Even in these cases, X contains no exceptional curve [copy of P’(C) with self-

intersection - 13, hence nor does S: S is always minimal.

PROPOSITION 5.1. Any complex surface (even non-compact) with geomerric srructure is

minimal.

PROPOSITION 5.2. If S has geometric structure of type P’(C), P’(C) x P’(C), P’(C) x 6: or

P’(C) x H, rhen b,(S) is even and x(S) = - 1.

For in these cases X is covered by embedded copies of P’(C). which all project down to

rational curves covering S: thus S is ruled.

We now investigate the converse question of which ruled surfaces have geometric

structure: we may suppose S minimal. Such surfaces have been classified (see, for example,

[3, p. 303): either S z Pr (C) or there isa map S + B onto some smooth curve B with each fibre

isomorphic to P,(C).

In this case, S is a PI (C)-bundle over B: since dim B = 1, this projective bundle is

associated [3, p. 291 to a vector bundle C’ + E + B. We now need to refer to the classification

of plane bundles over curves. However, if L is a line bundle, L @E and E have isomorphic

projective bundles.

If B has genus 0, B z PI(C), any plane bundle is the sum of two line bundles, and line

134 C. T. C. Wall

bundles are determined up to isomorphism by the Chern class, or degree, which we may

regard as an integer. Thus E = mH@nH [where c,(H) = I], and P(E) is determined up

to isomorphism by jm - nl. These are called Hirzeburch surfaces after [17]: only the case

m - n = 0, giving P’ (C) x Pi (c), has geometric structure. This is also [2] the only case when

the ruling is not unique.

For the case when B has genus 1, plane bundles were classified by Atiyah Cl]. Either we

have the direct sum of two line bundles or we have an irreducible bundle: but all of these only

define two distinct projective bundles A0 and Ai, according to the parity of cl. As to the

reducible cases, we may suppose that the line bundles have respective degrees 0 and d 2 0. According to [45], all those with fixed d > 0 (and a given elliptic base curve B) give isomorphic

complex surfaces Sd, while those with d = 0 give a family 9’, of surfaces parametrized by B itself (modulo automorphisms).

If S has a geometric structure, our bundle must be a flat bundle, induced by a

representation of 7ci (B) in PU2. As ret (B) is free abelian of rank 2, either this representation

lifts to U1, where it is reducible, or each generator interchanges the fixed points of the other on

P’Q), so the subgroup of PU2 is isomorphic to the four group. In fact it is well known [36].

[45] that each bundle with d = 0 admits a unique expression as a bundle induced from

u, x U,: we thus obtain all the surfaces in Y_. The flat bundle induced by sending the

generators to (the projective classes of) (: _b)and(_O A) can be identified (see, for

example, [45]) with Ai. The surfaces &(d 2 l), A0 do not admit geometric structures.

If B has genus g 2 2, plane bundles over B do not admit a simple classification, though a

great deal of work has been done on their moduli space. This has tended to focus on

Mumford’s [34] concept of stable bundle. The key result for us is the following, due to

Narasimhan and Seshadri [36].

THEOREM 5.3. A cector bundle of rank n and degree q, with 0 5 q < n, is stable ijand only if

it is associated to an “irreducible unitary representation of xl(B) of type T”.

It remains to explain the final term. In fact, we have a projective representation: if

(% 81, . , rg, 8,) denotes a standard system of generators of xi(B), we require

P(~i)P(B,)P(~;‘)P(B; ‘) . . . p(j?, ‘) = exp (- Zniq/n)l.

Since any vector bundle of rank n can be tensored with a line bundle to have degree 4 as

above, we deduce that if our surface S corresponds to a stable plane bundle, it is geometric.

Again, if we have a sum of two line bundles, each of degree 0, then we have a representation

in Ui x Ui, and hence a geometric surface. But a sum of two line bundles of different degrees

is not geometric in this sense (nor are the unstable bundles, which are irreducible, but

extensions of a line bundle of degree d, by one of degree d, > dl, so deform to such sums).

96. CHARACX’ERlSTlC NUMBERS

For any closed oriented Riemannian 4-manifold there are (see, for example, [27, Vol. 2,

Chap. 121) differential 4-forms o i, 02, defined in terms of the local geometry, whose integrals

give the characteristic numbers of M (signature and Euler characteristic):

00 = J,wml X(M) = j,p,.

Now the bundle of 4-forms has one-dimensional fibre, so at any point, or and oz are linearly

dependent. Thus if M is homogeneous, or at least locally homogeneous, a relation

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 135

ulol + u2wz = 0 which holds at some point holds everywhere, and thus implies

U,cJ(iM)fU~~(M) = 0.

In particular, if IV has a geometric structure modelled on X (say), the above applies with the

addendum that the linear relations (ai, a*) depend only on X. not on the particular .M.

Todetermine these relations, first observe that whenever G, is not semisimple we found in

92 a normal subgroup Vof G.isomorphic to a real vector space such that for any lattice F in

G,, F n Vis a lattice L in V. This induces a foliation of the manifold M = F \G,/ K,y by copies

of the torus L\ V. It follows from the fact that the tangent bundle of iti splits as a direct sum

that x(M) = 0: if the leaves have odd dimension we also obtain I = 0. The only cases

remaining are X = HZ x E*, when it is clear that 0 vanishes for some (hence for all)

corresponding M, and X = FS, which does not correspond to any compact M.

THEOREM 6.1. Let M 4 be a closed oriented geometric 4-manifold, modelled on X. Then the

characteristic numbers of M satisfy:

IfX = P2(C), H2(C): a(M) > 0, x(M) = 30(M).

IfX=S2xS2,H2xH2,S4: a(M) = 0, x(M) > 0.

IfX=S2xH2,H’: a(M) = 0, x(M) < 0.

0 therwise: a(M) = 0, x(M) = 0.

Proof. If X = P’(C), S* x S* or S4, we can take &f = X and compute the characteristic

numbers: by the above, any linear relation is inherited by all other oriented geometric

manifolds with the same model. The same argument also shows that the inequalities are

inherited.

If X = S2 x H ‘, H 2 x H 2 we argue similarly, but replacing H * by a closed surface of genus

g > 1 (which has model Hz).

For X = H2(C), H4 there are no very simple models. The result in these cases (which is in

any case well known) can be obtained by comparing the differential forms oi, wt for X with

those for the compact dual of X as symmetric space, viz. P2(C), S4, respectively.

We turn to the interpretation in the complex case. Let S be a compact complex surface,

homeomorphic to M as above. We have a dictionary of real and complex characteristic

numbers:

z(M) = et(S), a(M) = 3Cd(S)-2~2(S)l

to which we may add for completeness the Pontrjagin number

and the arithmetic genus ~1 (M) = 30(M)

x(fs) = Tt cc:(S) + c2Nl.

The following relations between these and the classification are known.

For a minimal complex surface S,

If k(S) = 2, then c:(S) > 0 (see, for example, [2, p. 2073).

If K(S) = 0 or 1, then c:(S) = 0 (see [2, p. 1881 or $47, 8 below). If s = P’C, c?(S) = 9.

If S is a P’-bundle over a surface of genus g, c:(S) = 8(1 -9).

If K(S) = - 1 and b,(S) is odd, then b,(S) = 1 (see 99)

so c2C3 = x(S) = b,(S);

136 C. T. C. Wall

also x(c,)= l-q+p,= l-1+0=0 [29,Th.26+Th.ll+Th.3]

SO c:(s) = -b2(S) IO.

To include non-minimal surfaces note that blowing up a point subtracts 1 from cf.

In particular, if cf (S) > 0 then either h.(S) = 2 or S is rational. If S has geometric structure

of type H2 (C) or H x H, it is clearly not rational (being aspherical), so has X(S) = 2. There is

no general structure theory for surfaces with h.(S) = 2, but a famous theorem of Yau 1501

(cf. also [33]) characterizes one of our classes.

THEOREM 6.2. [SO]. For all surfaces with IC = 2, c:(S) I 3c2(S). The equaiiry

c:(S) = 3c2(S) holds if and only if S has a geometric srructure of type H’(C).

In fact until quite recently relatively few examples of surfaces with c:(S) 2 Zcz(S)

[equivalently, a(S) 2 0] were known, but a spate of new ones have now been constructed (see,

for example, (191). At one stage it was conjectured that all such surfaces were aspherical: this

has been disproved by an example of Mandelbaum [31]. Theconjecture that all have infinite

fundamental group remains open (*), as does the problem of characterizing surfaces with

geometric structure of type H x H. To the best of my knowledge, no examples are known (*)

of surfaces S with x(S) = 2, a(S) = 0 which are not of this type. Certainly such surfaces need

not be (finitely covered by) products of curves: we can take I to be the group of units in a

quaternion algebra over a real quadratic field, split at both infinite primes. A partial

characterization is due to Enoki (private communication).

(*) Added in proof Examples of simplyconnected surfaces with cf > 2c,, and one example

with c: = 2c2, have now been discovered. See “Simply-connected algebraic surfaces of

positive index,” by B Moishezon & M Teicher (preprint, Columbia University).

97. ELLIPTIC SURFACES

S is called an elliptic surface if there is an analytic map y?: S --f B whose general fibre is an

elliptic curve. We will (as is customary) suppose that S is non-singular and that no fibre

contains an exceptional curve of the first kind: such S is called “relatively minimal” (in fact,

except when S is rational, this is equivalent to minimality in the usual sense).

A classification of possible fi bres of $ was given by Kodaira [28, 6.23: the cases are labelled

It(k 2 0), II, III, IV, I:(k 2 0), II*, III* and IV*. Case I, means that the fibre is a smooth

elliptic curve: all other types of fibre we shall call sing&r. By considering a neighbourhood of

a fibre, Kodaira also (lot. cit.) defines a multiplicity m 2 1: m > 1 only for finitely many fibres,

and these must all be of the type I,, for some k. A fibre is called mulriple if m 2 2 and

exceprional if it is either singular or multiple.

A non-exceptional fibre F,,(b EB) has an invariant j(Fb)EC. This extends [28, $71 to a

regular map j: B -+ P’ (C). We have j = 0 for fibres of types II, IV, II* and IV*; j = 1 for those

of types III and III*;j = Y, for those of types II, and I: with k 2 1: for types 1,and I:, j can

take any finite value. Kodaira calls j the functional invariant of the surface S.

Crucial for the study of elliptic surfaces is the following formula for the canonical divisor

[29, Theorem 121:

K,= $*(K,+D)+X(mi- l)fi

where the Fi (with multiplicities mi) are the multiple fibres and

degD = 1 -q-tp, = ir(f,)

is the arithmetic genus of S. A first deduction from the formula is that Kt = 0. Then by

Noether’s formula

GEOMETRIC STRUCTURES OS COMPACT COMPLEX ANALYTIC SURFACES 137

where 1 is the Euler characteristic of S, hence equal to the sum of those of the singular fibres,

which are given by Table 6.

Table 6

Type of F It II III IV I: II’ III’ IV.

X(F) k 2 3 4 k+6 10 9 8

In particular, x 2 0 and vanishes only if there are no singular fibres.

We often regard B as an orbifold (terminology of Thurston [47]) alias v-manifold

(terminology of Satake [41]), with a 2n;rni cone point at each point P, corresponding to a fibre

of multiplicity mi. Then the orbifold Euler characteristic of B is (by definition)

xorb(B) = 2-2g(B)-E(1 -m;‘).

The structure of Sat a smooth multiple fibre is homeomorphic to the product ofa circle with a

multiple fibre in a Seifert fibration of a 3-manifold.

If Fi is the fibre over Pi, then as divisor Il/*Pi = miFi. Thus if m is divisible by all the mi,

mK,= 1(1* 1

m(K,+D)+Em ( ‘!) l- l-m_ Pi - $*(D,)

is the pullback of a divisor D, of degree

m[Zg(B)-Z+X(cJ] +Em(l -miW’) = m[x((rs)--%o’b(B)] = mTS,

where T~ = x(lrs) - xorb(@.

LEMMA 7.1. The Kodaira dimension of S is given by

ti = sgn (rJ

i.e. h’ = - 1, 0 or 1 according as ?s < 0, TV = 0 or TV > 0.

Remark. This result appears to be well known, but I am unable to find it stated explicitly

in the literature (cf. [2, p. 1621).

Proof. If m is divisible by the m,. then

P,(S) = HOC&s; $*(D,)] = HO(B; D,),

and deg D, = mTS. If r5 < 0, we have negative degrees and these all vanish, hence so do all P,,,.

Thus K = - 1. If TV > 0, then for large enough m, by the Riemann-Roth theorem,

P,(S) = degD,+ 1 -g(B)

increases as a linear function of m, hence K = 1. (Note that by (4.0) to compute K it is enough to

know all the Pk,O for any particular m,.)

If rs = 0, then deg D, = 0, so P,,, = 0 or 1. It remains to show that the value 1 can occur.

Now ifg(B) 2 2, we have TV > 0. Ifg(B) = 1, then rs = Oonly ifthereare no exceptional fibres.

We return to this case below. Otherwise g(B) = 0: but then deg D, = 0 is enough to ensure

h’(B; D,) = 1.

For the final case, if bl (S) is odd, as bl (S) 2 bl (B) = 2, S is in Kodaira’s class VI [29,

Th. 261, so P, (S) = p,(S) 2 1. If b, (S) IS even, then if h.(S) = - 1, S is ruled. But any rational

curve on S admits only constant maps to B so lies in a fibre which, too. is elliptic: a

contradiction.

138 C. T. C. Wall

The importance of elliptic surfaces for the classification is shown by the first part of

LEMMA 7.2. (a) 1f~(S) = 1, or ifx(S) = 0 and bi (S) is odd. then S is elliptic. (b) In these

cases the elliptic structure is unique. (c) If S is elliptic and bI (S) is odd, then S has no singular

jibres.

Proof. (a) follows easily from the results of Kodaira. By [29, Theorem 573, if K(S) 2 0

then Pi2(S) > 0. Thus S is in one of the classes 2-6 of [29, Theorem 551. But classes 2 and 3

(K3 surfaces and tori) have K(S) = Oand b, (S)even; class 5 is the set ofsurfaces with K(S) = 2.

The remaining classes 4 and 6 consist of elliptic surfaces.

Part (a) can also be seen directly as follows. If K = 1, the pluricanonical map 4, has one-

dimensional fibres E. These satisfy E2 = 0 and K. E = 0 (since they are contracted by +,,,),

hence have genus 1. Similarly, if the function field of S has transcendence degree 1. a non-

trivial function defines a map from S to a curve, and the same argument shows that the fibres

have genus 1 ([29, Theorem 43).

For (c) we consider the fundamental group of S. If A is a finite subset of B containing the

images of all exceptional fibres, and F a general fibre, we have an exact sequence

{l)+n,(F)&JS-$-‘(A)]%ri(B-A)+{l}.

An examination of the neighbourhoods ofexceptional fibres shows that this leads to an exact

sequence

‘* *. ~1 (F) -+ n,(S) -+ ~1 Orb(B) --, { l},

where nprb(B) denotes the fundamental group of B as orbifold. If there are no singular

fibres, i, is injective; otherwise it is clearly not. More precisely, it can be seen [16]. [21, II]

that the image of i, is finite cyclic (in fact [Sl], it is trivial). Thus $ induces an isomorphism

H,(S; W) 2 HI (B; R), so in particular b,(S) = b,(B) is even.

AS to (b), if we have any elliptic surface S then (as we have seen above) any pluricanonical

map must collapse each fibre to a point. Thus if x(S) = 1, fibres of II/ can only (cf. [Zl, II]) be

the connected components of the fibres of any pluricanonical map with one-dimensional

image. If bl (S) is odd, then the field of meromorphic functions on S has transcendence degree

1 (not 2, since S is not algebraic; not 0, since S is elliptic) so if C is a model for this field, and

B 4 C the corresponding map, the map I(/ must factor through rr: in fact, the two must be

equivalent.

Observe that both (a) and (b) may fail for algebraic surfaces with h’ = 0: for example, K 3

surfaces need not be elliptic (though they deform to ones that are) and Enriques surfaces have

in general (hence K3 surfaces often) two distinct elliptic structures [2, 17.71; similarly for

complex tori.

It is time to return to our geometric structures.

PROPOSITION 7.3. (a) Any complex surface H.irh compatible geometric scrucutre of type

C x H, Nil x W or ST, x R is elliptic. (b) AnJ elliptic surface with geometric srructure

compatible with irs complex structure has no singularjbres.

Proof (a) S is the quotient by a discrete group lY (acting on the left) of one of:

LQ’ X SL,(R)jSOl. R x (Ni/3rxSOz)jS02, R x (R x zS^vL2): SOz.

In each case we know [by (2.3) or (2.6): see the discussion in $21 that r meets the centre of G,

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 139

(isomorphic to iw2 in each case) in a lattice L in W’. Moreover, this subgroup has Lie algebra

invariant under J in each case. Thus the elliptic curve L\,C acts on I’,,G,/SO, defining a

fibration, and almost all the fibres are isomorphic to L\,C itself.

(b) If S has a singular fibre. then this is a union of rational curves, each with negative self-

intersection number. But we have already seen that no surface with geometric structure carries

any such curves.

We insert parenthetically here a construction giving many such examples with 6i odd,

which the author found helpful in giving insight. Let (X, 0) be a normal surface singularity

with good C “-action (cf. [39]): choose L E C x with IE.1 > 1, and factor X - 10) by the

subgroup generated by L. The result is a compact surface with bi odd: the orbits of the C x-

action project to elliptic curves which fibre it. The base of the fibration is the orbifold which is

the quotient of X - {Oj by C x. This generalizes the original construction of Hopf [20]: the

present version is largely due to Neumann [3S].

We now consider the structure of our elliptic surfaces in greater depth. From Proposition

7.3 (and Lemma 7.2) we see that we need only study those without singular fibres. It follows

that j cannot take the value co, so is constant on B. We must now consider multiple fibres.

An orbifold covering map B + B induces an unramified covering s’ -+ S, with S’ elliptic

over B’, with multiple fibres corresponding to the cone points of B’. Now any orbifold B is

either [42]. [47]

good: i.e. admits a finite orbifold cover B’ with no cone points; its universal cover is then

isomorphic to P’(C), C or H; or

bad: when B has genus 0 and either one cone point or two, with different multiplicities.

For an elliptic surface S with no singular fibres corresponding to a bad orbifold B,

xorb(B) > 0, so 5, < 0, h’ = - 1.

Our objective is to prove the following, the case br odd of which is closely related to a

result due to Inoue: see [30].

THEOREM 7.4. An elliptic surface S without singularjbres has a geometric structure ifand

only if its base is a good orbifold. The type of this structure is determined as follows:

K -1 0 1

b , even c x P’(C) Cl CXH b, odd S’ x E’ Nit' x E' SIL*xE'

Our plan of proof is first to prove the existence of a geometric structure of the indicated

type, and then show that no structure of any other type is possible.

By hypothesis, B is a good orbifold; let B’ be a finite cover of B free of cone points. The

normalization S’ of the pullback is a bundle over B; its (topological) monodromy group acts

by analytic automorphisms on the fibre E, hence is finite [in fact, if it is non-trivial, the image

of 7c1 E + rrr S is finite so bI (S) is even]. Taking a further cover B”, we may suppose the pulled

back surface S” has trivial monodromy. We first consider this case.

As the holomorphic monodromy can now only consist of translations of the fibres, the

associated Jacobian fibration [28, $83 is a product:

JS = E x B.

According to Kodaira [28, 5 lo]. such elliptic surfaces S are classified by H1 (B; Q(JS)). As JS

140 C. T. C. Wall

is trivial, we have an exact coefficient sequence

0 + Y --* e,-+ Q(J.s) --+ 0,

where 3 is a trivial bundle with fibre the lattice L corresponding to E:

O+L-C-+E-0.

This gives an exact cohomology sequence

H’(B; Y)-Hl(B; P,)+H’(B; Q(JS))I,H2(B; -EP)+H2(B; f,).

In this sequence, HZ@; fiB) = 0 and H’(B; cB) = H’*‘(B) is a vector space over c of

dimension g(B). Also H2(B; 9) z L, and the image c(q) in L of the class q corresponding to S

gives the topological part of the classification.

As we have a fibre bundle with trivial monodromy, n,(S) has a presentation of the form

(ai, bi, . . , agl b,, c, dl c, d central; a;‘b;‘a,b, . a,‘b,‘a,b, = c’dS).

Here L g xl(E) is the free abelian group on c and d; we can identify the element c(q) above

with c’d’. It is thus clear that

if c(q) = 0, b,(S) = 2g + 2 is even

if c(q) + 0, b,(S) = 2g + 1 is odd.

We may, and in future will, choose the basis of L such that c(a) = cr for some r 2 0. Observe

also that with the current hypothesis, K(S) = K(B) is determined in the usual way by g(B).

We now consider the various cases.

K = 1, bl even. We consider the group acting on C x H generated by translations of C by

L, and the operators

Ai(W, 2) = (W+ ai, ‘ii(Z))

where the xi E PSL,(R) generate a discrete group acting on H with quotient B, and can be

normalized to satisfy

The quotient by such a group gives a fibration over B with fibre C/L = E. We appear to have

2g parameters ai here as opposed to the g coming from H ‘*O(B) above: this is explained as

follows.

Each holomorphic l-form OE H ‘*I B induces a form & on the universal cover B z H. ( ) Such an 65 is exact: say & = d4. We have

4(zi(z)) = 4(z)+ Pi

where the pi are the periods of w. The map

Q(W, Z) = (W + 4(Z), 2)

gives an automorphism of 43 x H transforming the group with translations by the ai to one

with translations by ai + pi. Thus sets of parameters a E HI (B; C) differing by an element of

Ho* ‘(B) give rise to isomorphic surfaces, and the image of a in H’*‘(B) is the effective

invariant.

This already shows that all surfaces of the type considered have a geometric structure

modelled on C x H. For the exact classification, consider the above map H1 (B; 9) -+ H’(B; C,). This is the composite of the inclusion H’(B; 9’) -+ H’(B; C) (isomorphic to

L2g c C29) with the projection of H’(B; C) on H I. O(B). The image is not in general discrete,

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 141

and the composite fails to be injective precisely when there exist non-constant maps I3 ---L E.

All this is reflected in the geometric structure: in particular, factoring out H’(B: Y) corresponds to the remark that only the values modulo L of the a, are significant.

K = 1, b, odd. We recall from $1 that the group acting on 63 x H here is generated by

translations in ,c together with the action of Szz(R) by

(; i;) (w,:) = (ir-210g(cz+d).~).

Thus to uniformize our surfaces we take the xias above, choose some lifts 5, to ST,(W), and set

Ai(w, --) = ~i(W, =) + (Ui, 0)

for some complex numbers 0,. Since ST, commutes with the translations, the commu-

tator product of the Ai coincides with that of the ii, which (of course) lies in the kernel of

Sx, -+ PSL2. It was calculated in [37]: the result is (2 - 29) times the generator, deter-

mining a translation of C by (2 - 2g)(2rri). Now since this corresponds to the Chern class of

the bundle, it must coincide with the element c(q) defined above. Since we normalized

c(q) = c’ (or additively: TC), the generator CE L must equal ~~‘(2 - 2g)(Zni)d. This

identification is no restriction: it simply determines a preferred embedding of L in 43. Now the

rest of the discussion is the same as in the case with b, even.

K = 0, bi even. In this case, the model space is C2: for the cases under discussion we only

need the translations.

Let ri, x2 E C generate a lattice corresponding to B: we generate a group acting on C 2 by

translating the first coordinate (w) by elements of L and by Ai (w, z) = (w + ai, z + ri). The

quotient gives a fibration over B with fibre E and monodromy of translations only: as in the

case K = 1, we see that we obtain all such fibrations.

K = 0, b, odd. Here again the model space is C ‘; there is a group bijective to C 2 which acts

by T,,, (w, -_) = (a + w - ib;, b + z).

If ri and x2 are as above, we again generate a group by translating \V by elements of L and by

T,,,,, and Ta2,z2: the quotient gives a fibration over B with fibre E and monodromy consisting

of translations. The first Chern class corresponds to the commutator of T,,,l, and Talrm2, which is translation of H’ by i(Giu2 - c2z1): twice the co-area of the lattice spanned by rl and

r2. As in the case ti = 1, provided we identify this vector with rc EL, our conclusions follow as

before.

K = - 1, b, even. When B has genus 0, the map c is an isomorphism. Thus for c = 0 we

have a unique surface B x E = P’ (C) x E, which is the quotient of P’ (C) x C by the usual

group L of translations.

K = - 1, b, odd. In this case our model space is C2 - { 0): there is a standard projection to

P’ (C), with fibres the punctured lines through 0. Choose i. such that C “/( j. ) is isomorphic to

E: additively, we can think of C modulo the lattice generated by 2rri and log i.. For suitable E.

we can identify this with L in such a way that 2ni corresponds to c E L. Then the quotient of d) 2

- (0) by (x, J) + (j.x, ;.I‘) gives a fibration over P’ (C) with fibre E and Chern class c. To

obtain Chern class rc, we factor out the group generated by

(x, y) + (;.‘/I x, ;.‘I’ x) and (x, y) -+ (e21iirx, e2nii’y).

This concludes the list of cases.

We now return to the general case contemplated at the beginning of the proof.

132 C. T. C. Wall

We have constructed a diagram

Y-, B”-,B

where TC, TC” are elliptic fibrations with fibre E; rcx has fibre C; 71” has no multiple fibres and

monodromy consisting only of translations; S” is the quotient of X by a (cocompact) lattice I-”

in G,, and is a finite covering of S, which we may clearly suppose regular. Lifting this, we

obtain a group r acting on X with quotient S, and containing I-” as normal subgroup with

finite index. We will show that r c G,, so preserves the geometric structure.

By construction, each covering transformation of S” over S preserves the elliptic structure.

We first consider the base. Here r, a finite extension of r”, acts properly and holomorphically

on Y with compact quotient B. If Y = H (K = l), it follows that this action is by a discrete

cccompact subgroup of PSL,. If Y = C (K = 0), r is contained in the group of automorphisms

of C, which is the affine group. As it has a subgroup I-” of finite index, of translations, the

projection to GL, (C) is finite, hence lies in the circle group, S02. If Y = P’ (C) (K = - I), r” acts trivially, so I- is finite, hence conjugate to a subgroup of PU, = S03.

Next we consider the effect on a fibre. Each fibre of S” is isomorphic to E; the effect of a

covering transformation over S coincides (module such identifications) with an automorph-

ism of E. Such an automorphism lifts to one of C of the form

W-+&Wfll

where in general E’ = 1. If E is harmonic (j = 0), s4 = 1; if E is equianharmonic (j = l),

t? = 1; thus in any case E l2 = 1 Since the monodromy (topological) in S” is trivial, or the .

analytic one consists only of translations, the element E corresponding to a given covering

transformation cannot vary from fibre to fibre, but is constant.

Suppose bi even: then X = C x Y. For each g E I’ we have seen that g defines a geometric

automorphism zg of Y: from this and the effect on fibres we see that

g(w, 2) = (EW + a(:), Q(L)).

It remains to show that a(z) is constant. But g normalizes the group generated by

Ai(w, Z) = (W + Uiy s(i(Z)),

where the Zi generate a discrete group I=” acting on Y with compact quotient. Since

CJelAig(W, i) = (W +&-‘Uif E-l U(Z)-U(Z,’ CliCt,g(Z)), Ctl’OIiCtg(Z)),

E-‘U(Z)--u(r,’ Cti a,(Z)) is constant. NOW ifs = 1, we deduce that a is constant on the orbits of - I, r = ,x ; ’ I=” ctg, and hence factors through the quotient Y/F” = B”, which is compact. Thus

in this case u(z) is constant. If E # 1, the invariance condition defines a line bundle of finite

order (that of s) over B”, for which u(z) defines a section. But since this bundle has degree zero

and is non-trivial, the only holomorphic section is 0, so in this case u(z) = 0. This concludes the

proof when b, is even.

If bi is odd but K = 0 or 1, we only need slight modifications to the above. Again X Z

C x Y and the action on Y is as before. However, in these cases for each z acting on Y there is a

function /, (2) on Y such that the geometric automorphisms are the

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 143

The invariance of the Chern class shows that the automorphisms of the fibres must satisfy

E = 1. We can thus write our chosen g E Ias

(w, z) + (w + a(-_) + I,(:), $2))

and it normalizes a group of elements of the form

B: (w, Z) -t (\c. + b + /,(:). B(Z),.

Since r + 1, satisfies a cocycle condition, we deduce on computing g- ‘Bg that if this belongs

to I”, then a(z) - a(~- ‘/3x(z)) is constant. The proof now concludes as before.

Finally, in the case when b, is odd and k = - 1, I induces a finite group acting on P’ (C),

which we may take as geometric. I claim that any lift 3 of such an element to an automorphism

of C* - {Oj is linear: it suffices to verify this for the identity element. On each fibre C ’ this

induces an automorphism which must be multiplication by some non-zero constant a

(interchange of the ends is ruled out since a preserves the Chern class of the fibration). NOW

a(z) depends holomorphically on z E P’ (C), hence is constant. This concludes the proof of

Theorem 7.4.

COROLLARY 7.5. Suppose S is an elliptic surface without singularjibres. Then the unicersal

cover ofS is biholomorphic to

CXH if h.(S) = 1

cz if X(S) = 0

CXP’C if x-(S) = - 1, b,(S) euen

C2- (0; af X(S) = - 1, b,(S) odd.

If the base B is a good orbifold, this is immediate; otherwise, if x = - 1 and b, is odd, we can

appeal to [29, Theorem 28]-where a significant part of this corollary is proved; if h’ = - 1

and b, is even, by the classification, S is ruled over a curve with genus 1.

$8. TRIVIAL PLURICAiNONICAL BUNDLES

The following is an easy consequence of the main Kodaira classification.

THEOREM 8.1. The following conditions on the compact complex surface S are equitalent:

(i) h.(S) = 0.

(ii) For some n, nK is a trivial bundle.

(iii) For somefinite unramijed cocering surface S of S, K(S) is tricial.

(iv) For some finite covering s’ of S, s’ is either a complex torus, a K3 surface or a Kodaira

surface.

Proof. (i) =- (iv) by the classification. (iv) =- (iii) by [29, Theorem 191: indeed, Kodaira

shows there that these are the only three types of surface with trivial canonical bundle.

(iii) =. (ii) for we may suppose Sa regular covering of S, and choose a non-zero section G. Each

covering transformation g then has ga = a~ where a is a holomorphic on 3, nowhere zero,

hence constant. This defines a homomorphism from the covering group G (which is finite) to

43 ‘: the image is cyclic, of order n, say. Thus 0 projects to an n-valued section of K: this

determines a nowhere zero section of nK. (ii) =-(i) for we have Pkn = 1 for all kf H: as

PI, 5 Pk,, every P, is 0 or 1.

144 C. T. C. Wall

A direct proof of(i) => (iii) is non-trivial: indeed, it is a significant step in the one version of

the proof of the classification theorem.

The three classes of surfaces in (iv) are quite distinct. since cZ > 0 for K3 surfaces and

c2 = 0 for the others; b, is odd for Kodaira surfaces and even for the others: these differences

are inherited by S from s’. By Lemma 7.2, surfaces with k = 0 and b, odd are elliptic, and so by

Theorem 7.4 geometric: they were fully discussed in $7.

The K3 surfaces are simply connected, and are certainly not geometric in the sense of this

paper. If S is covered by a K3 surface, S is an Enriques surface.

There remains the case when sis a complex torus, i.e. the quotient of C:’ by a lattice. Such s’

are clearly geometric. Surfaces S (not themselves tori) finitely covered by a torus are called

hyperelliptic. They were classified long ago (see, for instance, [2, p. 1481) and are all elliptic-

hence (or, better, by direct construction) geometric.

This completes the discussion of geometric structures.

Remark. Triviality of nK (for some n) in Pit (S) should not be confused with triviality of

the characteristic class ci in H’(S; W) (which it clearly implies).

If b, is even, the conditions are equivalent. For as ci = 0, ci = 0: thus I\’ < 1. If x = 0, some

nK are trivial; otherwise if S is elliptic, rs # 0 and ci equals rX times the class of a fibre, which is

non-zero in cohomology. If h: = - 1, S is ruled, and ci is non-zero on each curve of the ruling.

If b, is odd, ci = 0 for any elliptic surface, in particular for all minimal surfaces with IC = G

or ,Y = 1; while for h’ = - 1,

ci = 0 =-c: = 0 -x(S) = 0 q&(S) = 0 =>ci = 0

The result on elliptic surfaces holds since here the spectral sequence of the fibration is non-

trivial [as bl (S) # b, (B)]; the differential kills the cohomology class of the fibre, and cl is a

multiple of this.

As affine structures play an important role in the literature on uniformization. we insert

here some further references. The closest to our approach is the paper [25], where the

following are proved (in our terminology).

PROPOSITION 8.2. If M is a compact complex surface then:

(i) if M has a holomorphic ajine connection then c1 (M) = c2 (M) = 0 and M is minimal;

(ii) if bl (M) is even, M as in (i), M is either a complex torus or hyperelliptic; (iii) if b,(M) is odd, M as in (i), then either K(M) = 0 or 1 or K(M) = - 1 and b,(LW) = 0;

(iv) the converse holds (exceptfor certain Hopfsurfaces) and indeed such M have a holomorphic ajine structure.

The various possibilities for Hopf surfaces are also discussed in detail in [25]. We recognize

these affine structures as subordinate to our geometric structures C2, ST2 x E’, ,Vi13 x E’,

S2 x E’, SoIt and Sol’: (see also $9 below).

Affine connexions play a key role in the paper [5] to which we refer in 99. Further useful

references are [ 151, [46]: the former considers also other structures, in particular a projective

connection: a necessary condition in this case is cf = 3c2, which we encountered also in 96.

99. SURFACES OF CLASS VII

It is customary, following Kodaira, to call surfaces with x = - 1 and bI odd “ofclass VII”.

This is not precisely the terminology introduced in [29, I] (though Kodaira’s later version in

[29, IV] is equivalent to it), but is (cf. [2, p. 1883) a more convenient usage. The class contains

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 145

the significant subclass of Hopfsurfaces, which are those with universal cover C* - (01 (all

Hopf surfaces are of class VII by [29, Theorem 301). It was shown by Kodaira that for S of

class VII, S is elliptic if and only if there are non-constant meromorphic functions on S [29,

Theorem 41, and that all these elliptic surfaces are Hopf [29, Theorem 281: thus we first

discuss Hopf surfaces.

Kodaira calls a Hopf surface primary if its fundamental group is (infinite) cyclic: any Hopf

surface S has a finite unramified cover s’which is primary, so we next discuss these. According

to [29, Theorem 301. we have the quotient of C2 - IO; by one of

(a) T(z 1, z2) = (ZiZi, Zl-_?). 0 < ir,l I 1221 < 1;

(b) T(ri, Z2) = ($21 + i.rT,r2i2), 0 < 1221 < 1, i. + 0.

By [29, Theorem 311, a primary Hopf surface is elliptic if and only if it is of type (a) with

z{ = a\ for some positive integers k and 1.

We first discuss the non-elliptic cases. By [29, Theorem 323 we then have 7c1 (S) 2 H x Z/l

(for 1 2 1) and the covering group is generated by T [as in (a) or (b) above] and ci, where

U(Z1 ,i2)= (&li1,&ZZz), &\ = E: = 1,

in case (b) we also need s1 = ~7. The surfaces of type (a) contain two elliptic curves Ci (the

image of the zi-axis); those of type (b) only have the curve Ci: there are no further curves on

these surfaces (cf. [3.5]).

The surface S is geometric if and only if it has type (a) with la,1 = 1~~1; thus not all

geometric surfaces are elliptic.

In the elliptic case, the elliptic structure is unique by Lemma 7.2 and defines a quotient

orbifold B. We have shown in Theorem 7.4 that B is good if and only if S (or s’) is geometric:

the condition for this is that we have type (a) with [zi/ = Ir21, and hence k = 1. Otherwise,

1: /& defines the projection of an elliptic fibration of s’ with C,, C2 as the only multiple fibres.

The surface S is obtained from 9 by factoring out an appropriate finite group of

transformations, cyclic unless SL 1 = ct2. The classification of topological types of these elliptic

cases is (cf. Kato [26]) the same as that of Seifert fibre structures on S3.

For surfaces S of class VII which are not Hopf and hence admit no non-constant functions

we have the general results [29, Theorem 111 that 4 = 1 and 6r = 1: thus 0 = x(C,), so

0 = c: + c2 and c2 = z = b2. Since c: + c2 = 0, x = -c: thus the quadratic form on HZ (S; R) is negative definite. Other results are rather incomplete. An excellent summary of known

results on the classification of surfaces S containing curves is given by Nakamura [35]; there

are four classes which can be explicitly described: for the rest, the curves form a single cycle C

of rational curves, with C’ < 0, which do not span Hz(S; W).

Of more immediate interest to us are the surfaces with b2 = 0. Non-trivial examples

(denoted S,, S,) were described by Inoue [24], who proved a characterization theorem in

[23]. A paper by Bogomolov [S], [6] ex:ends this result to show that there are no further

surfaces with b2 = 0. (However, I understand that there are doubts whether the arguments in

[6] are complete.) We shall need this result in $10, and will indicate the results which depend

on it by “(mod B)“.

We now describe these surfaces. First, let M cSL3 (Z) have eigenvalues a, 8, fi with zz > 1,

/3 # B. Choose a real eigenvector (ai, uz, a3) belonging to 2 and an eigenvector (b,, b2, b3) belonging to B. Now let G,w be the group of automorphisms of w x C generated by

go (WY -1 = cm, B4

gi OL., Z) = (bv +Ui, z + bi) i = 1, 2, 3

146 C. T. C. Wall

and define S,ti to be the quotient of H x % by G,V. It is easy to see that G,M acts properly

discontinuously and that 5, is compact. We see by inspection that (with the notation of 93)

gi E SO~: and that the gi generate a discrete cocompact subgroup of SoI4,. Conversely, by

Theorem 3.2, for any lattice r in Soi:, r meets ~~ in a lattice L in R3 so T/L is a discrete

cocompact subgroup of the quotient, W x SO1. Now if this intersects SO2 non-trivially, L is

invariant under an element of finite order. Thus L contains a fixed vector v by this element.

However, such a v is an eigenvector of R, so we have a contradiction (a one-dimensionai lattice

cannot have an automorphism of infinite order.)

Thus T/L is cyclic: also it cannot intersect the factor W non-trivially. else L wouid admit an

automorphism with real eigenvalues, of which two are equal. Hence all three are rational, and

integral (since L is a lattice), contradicting the fact rhat the automorphism has infinite order.

Thus the generatorg of r/L has no real eigenvalues, and is represented on L by a matrix M

as above. It now follows that the action of r on H x C is as described above.

Inoue’s second family of examples has two subfamilies as follows.

S,:. Let N sSLt (Z) have real eigenvalues 2, zL- ’ with corresponding real eigenvectors

(a,, az). (b,, b2). Choose a non-zero integer r, a complex number t, and further real numbers

cl, c2 satisfying an integrality condition to be made precise. Define automorphisms of H x C

by

go (w, -_) = (ciw, z + t)

gi(W,Z)= (w+ai,z+biW+Ci) (i = 1,2)

gx(w,:) = (w,z+r-‘(bla2-b2al))

Then these generate a discrete group acting freely, whose quotient we denote by Sz.

S,; is defined by modifying the above as follows. NE GL, (Z) has real eigenvalues

z > 1, -z- ‘. The rest are as above, except that we do not choose a t, but define instead

go(w, z) = (aw, -2).

We interpret these in terms of geometric structures of types Sol;, Sol’:: the latter

occurring in the case t $ W: we also recall that the full group of holomorphic automorphisms in

the case of Sol’: has two components. We defined Soi’: as a group of matrices, and the action

by matrix multiplication

for ?E E. each of the above gi belongs to this group.

To show that all discrete cocompact subgroups rare of the form above, let us write Go for

the connected component (E = 1) of Sol:, G, for its derived group (E = I = I) and G2 for the

derived group of G1 (E = z = 1, a = b = 0), and set Ti = F nGi. By (2.2), rl is a lattice in

G1, and by (2.3), Tt is a lattice in Gz. Thus rl/r2 is a lattice in G1/G2 z R’, and is invariant

under the generator S,, of To/T1. If i. -. 2 E W ‘; l-,/T2 has generators S1 = (al, b,) and

J2 = (uz, bz), and N is the matrix (with respect to this basis) of the automorphism of rl/r2

induced by do then we recover the above structure of rdr, (conjugating in Go/G?. we may

easily arrange a0 = b. = 0).

Nexl,g;‘g;‘g,g, isa non-zeroelement ofr,,so must besomepowerg; ofthegenerator

g3 of r2: hence g3 is as above. And g; ‘g;‘gOg;‘~g~~*also belongs to r2 (i = 1,2), so must be a

power ofg3: this yields the integraliry conditions on c1 and c2 referred to above. We have thus

recovered the above normal form for Si.

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 147

If F meets the non-identity component of So/:, we first assert that T/l-i is still cyclic. For

suppose not: then F contains an element with E = - I, 2 = + 1. Now if a = 0 this element has

order 2, so does not act freely; if a # 0, its square defines a non-zero element of Ii; TZ which is

an eigenvalue of the action of T,,/T, and (compare the So/l, case above) this yields a

contradiction. Now we can conjugate any element (with E = - 1, 2’ # 1) into the normal

form with a = b = c = 0. The rest of the argument is as before.

For the remaining cases recall that Sol; was interpreted as acting on C x H by

replacing the term i log 3 by im log r for any m # 0 gives an equivalent action. Since we have

the same group, the discrete subgroups are as before; gi, g2 and g3 are unchanged, but g,, now

acts by

ge(w,z)= (CzH’,z+t+imlogrr) t,mER;

but then t + im log a gives an arbitrary complex number.

To summarize the above discussion, we have shown

PROPOSITION 9.1. The Inoue surfaces are precisely the complex analytic surfaces with

geometric structure of type

S M SOlZ

S: (t real), S,; Sol?

S; (t@ W) Sol’:.

0 10. TOPOLOGY AND GEOMETRIC STRUCTURE

In the previous sections of this paper, although topology has played a role, we have

concentrated on complex analytic structure. In this section we will show that both the

geometric and the analytic structures described earlier are determined (up to equivalence) by

the topology. More precisely, we have:

THEOREM 10.1. If Mt (i = 1,2) is a closed 4-manifold with geometric structure of type

Xi (i = 1, 2), and MI, M, are homotopy equivalent, then X1 = X2.

THEOREM 10.2. (mod B). If M4 is a closed 4-manifold with geometric structure of type X;

and S is a compact complex surface, homotopy equivalent to M, then the triple (X; n(S), b,(S)

(mod 2)) appears in Table 4 (Theorem 4.5).

We begin the proof of these theorems with some general remarks. First, we may replace

the manifolds by appropriate finite covers as and when necessary: the homotopy equivalences

lift, and we do not lose the information needed for the conclusion. It is thus enough to

consider a manifold M (or manifolds M,, M *) given as r\ X where r c G i: in particular, we

may suppose M orientable.

Next, the homotopy-theoretic information we will use is of three types: the homotopy

type of the universal cover fi (namely X); the characteristic numbers a(M) and x(M); and the

fundamental group rcr (M): our discussion accordingly in divided into several stages. We will

treat both theorems together.

148 C. T. C. Wall

First case: &f compact

Then X is one of the geometries P’C, S’ x S2 and SJ. The condition defining this case (as

likewise with the other cases below) is invariant under homoropy equivalence of M, so for

Theorem 10.1 we have both Mi and Mz as above. Since the three cases for X = fi are

distinguished by their second Betti numbers, the conclusion is here immediate.

As to Theorem 10.2, we first observe that since no manifold homotopy equivalent to S’

has an almost complex structure, the case X = Sf cannot arise. Otherwise we have S

homotopy equivalent of PC or to PC x P’C. The result in these cases follows from [2,

pp. 135, 2021.

Before turning to the remaining cases, we first observe that in all of them (cf. end of $2) X

is homeomorphic to an open set in RS, with finitely generated homology. Now if M’ contains

a smoothly embedded 2-sphere with self-intersection number ( = normal Euler class) _+ 1, this

sphere has a neighbourhood N with boundary a 3-sphere and M is a connected sum of P’Q)

with another manifold Mi. If also M has infinite fundamental group, its universal cover

contains infinitely many copies of N, so Hz (&‘) is not finitely generated. If M’ is homotopy

equivalent to M, the same follows for M’.

Thus if M is geometric, and is homotopy equivalent to a complex surface S which is nor

minimal (so contains such a 2-sphere) we obtain a contradiction, since in all remaining cases,

as ti is non-compact, n, (M) is infinite. Thus for the rest of the proof of Theorem 10.2, we may

suppose S minimal.

Second case: iii not compact, c7(M) f 0

The only geometry arising here H’C, so again Theorem 10.1 is immediate. For Theorem

10.2 recall that the characteristic numbers satisfy a(M) > 0, x(M) = 30(M), so if the

homotopy equivalence of S on ,M preserves orientation, c:(S) = 3cz(S) > 0 and as c: > 0

[and S is not rational, as ret (S) is infinite] we have h-(S) = 2 as required. If-as seems unlikely

to be possible-the homotopy equivalence reverses the orientation, then cf (S) = c2 (S) > 0

and we still conclude K(S) = 2.

Third case: fi non-compact, a(M) = 0, z(M) # 0

We have three relevant geometries here: S* x Hz, Hz x HZ and H4. For Theorem 10.1,

observe that in the first of these cases, fi is not contractible (as it is in the other two), while

x(M) > 0 in the H2 x H2 case and x(M) < 0 in the H4 case.

As to Theorem 10.2, we find on checking the table giving the classification of minimal

compact complex surfaces that c: = 2c2 # 0 occurs only if K = 2 (and here c: = 2c2 > 0)

or K = - 1 and b, is even, when we have a ruled surface over a curve of genus 0 (in which

case s’ is compact) or genus g 2 2, in which case c : = 2c, < 0. Thus if M is geometric of type

H2 x H” and is homotopy equivalent to S, c2 (S) > 0, so K(S) = 2: if M is of type S’ x H2 or

H4 then c2 (S) < 0, so K(S) = 0, b, (S) is even, and S is ruled: in particular, the universal covers’

is not contractible. This gives a contradiction in the H4 case, which concludes the proof.

Fourth case: h? non-compact and non-contractible; a(M) = x(M) = 0

Here we have just two geometries: S2 x E2 and S3 x E’. As the two spaces X have different

Betti numbers, Theorem 10.1 is again immediate.

Thus suppose (for Theorem 10.2) S a complex surface with c:(S) = c2 (S) = 0 and s‘ non-

contractible. Then K(S) = 2 contradicts ct (S) = 0. If h-(S) = 0 or 1, S is (or at least deforms to)

an elliptic surface: since c2 (S) = 0 this has no singular fibres. But then by Theorem 7.4 it is

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES 149

geometric, with universal cover isomorphic to C’ or C x H, and hence contractible: a

contradiction. Thus K(S) = - 1. Now if bl (S) is even, we conclude that S is a ruled surface

over an elliptic curve, whence the universal cover 3 is homeomorphic to S2 x EZ.

If bl (S) is odd, and hence equal to 1, we have b2 (S) = 1 (S) = 0. By Bogomolov’s resulr [6]

since s’ is not contractible. S is a Hopf surface, so its universal cover is homeomorphic to

S3 x E’. (We can avoid this argument: see below.)

There still remain nine geometries. viz. E’, Nil3 x E’, Nil’, Sol;,,, Sol:, Sol:, HZ x E*,

S1, x E’ and H’ x E’. The argument in this case follows a different pattern from the

preceding ones. First we have

LEMMA 10.3. (mod B). If S is a compact complex surface which is aspherical and has

cf (S) = cl(S) = 0, then S is geometric.

Proof. We have seen above that S must be minimal. As cf (S) = 0, K(S) 5 1. If K(S) = 1 or

if K(S) = 0 and bl (S) is odd. then S is elliptic and [since c2 (S) = 0] has no singular fibres: by

Theorem 7.4, S is geometric. If K(S) = 0 and bl (S) is even, then since [as c2 (S) = O] S is not a

K3 or Enriques surface, it must be a complex torus or hyperelliptic: again S is geometric. If

K(S) = - 1 and b,(S) is even, S is ruled and we deduce that 3 is not contractible, a

contradiction. Finally if K(S) = - 1 and b,(S) is odd, then S is not a Hopf surface (since s’ is

contractible), and b2 (S) = cl(S) = 0. Thus by Bogomolov’s theorem, S is an Inoue surface and

hence geometric.

We deduce from the lemma that Theorem 10.2 in the remaining cases will follow from

Theorem 10.1. It thus suffices to prove the following.

PROPOSITION40.4. Let X,, X2 be geometriesfrom the list E4. Nil3 x E’, Ni14, Sol:,,. Sol:,

Sol;‘, H* x Et, SL, x E’, H’ x El. Let Ti be a discrete cocompoct subgroup of Gxi (i = 1, 2).

If ITI is isomorphic to r2. rhen X1 = X2.

Proof. As before, we may pass to a convenient subgroup of finite index. We will, in

particular, suppose that Ti c G:i. Setting Mi = ri\ Xi we have l-i = x1 (Mi), SO bl (ML) is the

torsion-free rank of r,/[r,,r,] = rfb: we define this to be b, (l-i).

The last three cases are distinguished from the others on the list by the fact that only in

these cases is r not solvable: we consider these first. By (2.6), in the cases HZ x E*, Sx2 x E’, r

has a normal subgroup which is free abelian of rank 2: the quotient is a Fuchsian group. For

H3 x E’ there is a normal subgroup Z of translations: the quotient r is a lattice in SO;,,.

This cannot have a normal abelian subgroup, since any such subgroup fixes a (unique) point in

H3 or on its ideal boundary, which would then have to be fixed by l=, contradicting

cocompactness. Thus in the H3 x E’ case, r does not have a normal subgroup which is free

abelian of rank 2. We can distinguish the other two cases since we know that if I- acts freely

on H* x E* resp. ST, x E’, 6, (I-) is even resp. odd (and we achieve free actions by passing to a

suitable subgroup of finite index).

In all the remaining cases except Sol: we saw in 92 that r contains a normal subgroup To

which is free abelian of rank 3: and indeed in each of these cases we may suppose (passing to a

subgroup of finite index) that the quotient is a lattice in R, hence infinite cyclic. For the case of

Sol;’ we gave a detailed description of the possible groups Ti in 99: if y does not belong to the

derived group G1, its centralizer is only two-dimensional; G1 itself is three-dimensional but

not abelian. Thus there is no normal subgroup here which is free abelian of rank 3.

150 C. T. C. Wall

To distinguish the remaining cases it will suffice to consider the monodromy of the

corresponding extension of Z3 by Z, provided we allow for (i) any non-uniqueness of the

normal subgroup isomorphic to Z3, and (ii) use only properties which are not lost on passing

to subgroups of finite index. In fact, by the descriptions of the various geometries we already

know that the eigenvalues of the monodromy are as follows:

sois,: all distinct: one real and two complex

sol;.,: all distinct and rea1

,I?; Nil3 x E’: Ni14: all equal to 1: these cases are distinguished by the sizes of the Jordan

blocks which are (1, 1, I), (1, 2) and (3) in the three respective cases.

These distinctions are indeed clearly preserved on passing to subgroups of finite index-in

fact, we have already so passed to reduce from quasi-unipotent to unipotent elements in the

three final cases. In the SO/~, and Sol& cases, the centralizer of an element not in Z’ has rank

< 3; thus this subgroup is uniquely determined. Although this is not so in the other cases, it is

easy to see that a different choice of Z3 leads to an equivalent situation. More directly, these

cases are distinguished by the length of the lower central series.

This concludes the proof of Proposition 10.4, and hence of Theorems 10.1 and 10.2. The

least satisfactory feature of the proofis the dependence on Bogomolov’s result: in fact, we can

avoid the use of this in some cases. In the proof of Theorem 10.2, this was used only to show

the non-existence of surfaces of class VII, homotopy equivalent to a geometric $f” with

a(M) = l(.M) = 0. We can conclude more simply if we already know that for M4-or indeed

for some finite covering MT--b, (M,) # 1. This argument excludes geometries of all the

Kahler types X; it also excludes Nil4 and Sol3 x E’ (one of the So/:.,), where bi = 2: ,Vi13 x E’

where (for a subgroup of finite index), b, = 3; and ST, x E’ where b, can be made large by

passing to a subgroup of finite index. There remain the cases So/i., and H3 x E ‘. In the case of

H3 x E’, the general conjecture that any closed hyperbolic 3-manifold has a finite cover

which is Haken would imply here that by passing to a finite cover we can achieve bl 2 2.

Weclose this section by observing that although Theorems 10.1 and 10.2 are formulated in

general terms, the fact that the conclusions are invariant under taking finite covers makes

them rather selective in application. Indeed, we saw in $3 that though G,/G”,is non-trivial in

many cases, hardly any of these yield holomorphic automorphisms. There are thus many

geometric 4-manifolds of these types which (even if orientable) admit no complex structure.

The most striking example is E4: although there are only [2, p. 1481 8 possible

isomorphism classes of fundamental group in the complex case, there are [7] 75 possible in

the real case.

$11. GEOMETRIC STRUCTURES ON NOT-COMPACT MANIFOLDS

It is not my intention in this section to be in any way systematic: merely to describe some

interesting examples and perhaps signal the fact that there may be more to be discovered in

this area.

Up to now, we have discussed almost exclusively discrete cocompact subgroups I of G,

which act freely on X, thus giving rise to compact quotients. A first generalization is to allow I

to have fixed points on X. In the quotient S = T/X we can either leave the resulting quotient

singularities or resolve them to obtain a smooth surface s‘. An interesting case is the familiar

one with X = C2 where we factor out by a lattice to obtain a complex torus and also by the

map x + -_x, to obtain a Kummer surface, which resolves to a K3 surface. Again, some

(perhaps all) elliptic surfaces with singular fibres but with j constant arise by resolving

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES I5 1

singularities in a singular geometric such surface. If fixed points are allowed, there are plenty

of finite group actions on P’C. The quotients here are all unirational (hence rational), and

include the weighted projective spaces (cf. [9]).

A second generalization (which may ofcourse be combined with the above) is to allow I- to

be a non-cocompact lattice. The quotient then has finite covolume: in the cases with complex

structure it seems to follow that there is a natural compactification as a complete surface

(where again we may resolve the singularities). A famous class of examples here is that of

Hilbert modular surfaces, arising as arithmetic quotients of H’ x Hz: see [18] and other

papers of Hirzebruch. Such arithmetic quotients can also be obtained for H’(C): see [19] and

papers of Hofzapfel.

In this connection the geometry F” (omitted in most of the above) comes again into its

own. Recall that here GO, = iw2 =SL,(R); X = G i/SO2 is thus fibred over H with fibre

W2 = C. If we first factor out Z2, we have an elliptic fibration over H, and this is such that the

fibre F, over 5 E H hasj-invariant preciselyj (T). Now if I- is a lattice meeting W’ in Z2, r\ X is an

elliptic surface over l=\H [where I’= = T/Z’ is the image of r in SL,(iw)], but there will be

exceptional fibres corresponding to the fixed points (if any) of I= on H. More precisely, let us

regard j as taking values in an orbifold C, with invariants (2,3, ZD): P’ C is punctured at x, and

has cone points with angles rc at j = 1 and 2n/3 at j = 0. Then we have a commutative diagram

H d=, T

\I” J

HII=

Here H is the universal orbifold cover of C,: thus iis also an orbifold covering.

Conversely, if we have a (compact) elliptic surface S such that when the singular fibres with

j = x are removed the resulting open surface So has no singular fibres, and j/S0 is an

orbifold covering of C,, then S’has a geometric structure of type F4. For if B, B’are the base

curves of S, S 9 the universal orbifold cover of B” must be H, and the elliptic surface over B”

pulls back to one over H which we can identify with Z2\ F4. Now we can argue as in the proof

of Theorem 7.4 to show that the covering transformations over S ‘must lift to self-maps of F4 induced by GO,.

There is a close connection of the surfaces just discussed with the “elliptic modular

surfaces” studied by Shioda [43].

The constructions discussed above seem in practice to give rise to projective algebraic

surfaces. Our final examples, which do not fit so easily into the scheme of things, are non-

Kshlerian. These are the Inoue-Hirzebruch surfaces (cf. [lS], [24, II]) where we take a group

M x V (M a module in a real quadratic field K; Va group of unitskwhich acts geometrically

on H x H, but not with finite covolume-let it act on H x C instead, and compactify the

quotient by a cycle ofrational curves at each end. Other examples discussed in the survey [35]

(half Inoue surfaces, parabolic Inoue surfaces, exceptional compactifications) seem to be of a

similar nature, bllt using in some cases the geometry Nil3 x E’

Acknowledgements-Thanks are due to the University of Maryland (College Park) for providing the stimulus for me

to undertake this investigation, to the Max-Planck Institut (Bonn) for providing ideal conditions for the final writing-

up, and to SERC for financial support allowing me the time to do the research.

REFERENCES

1. M. F. ATIYAH: Vector bundles over an elliptic curve, Proc. London .Math. Sot. 7 (1957). 414-452.

2. W. BARTH, C. PETERS and A. VAN de VAN: Compacr Complex Surfaces. Springer, Berlin. 1984.

152 C. T. C. Wall

3. A. BEAIXLLE: Complex Algebraic Surfaces (transl. by R. BARLOW), London Math. Sot. lecture notes No. 68,

Cambridge University Press, 1983. 4. A. B~~J~ILLE: Surfaces K3 (609 in Sem Bourbaki 1982-83). Asterisque 105-6 (1983). 217-230.

5. F. A. BOGOMOLOV: Classification of surfaces of class VII with b2 = 0. Morh. USSR-Ix 10 (1976). 255-269.

6. F. A. BOGO.HOLOV: Surfaces of class VII, and affine geometry, Moth. USSR-1:~. 21 (1983), 31-73.

7. E. CALABI: Closed, locally eudidean, 4dimensional manifolds, Bull. Am. Math. Sot. 63 (1957). 135, 8. W. L. CHOW and K. KODAIRA: On analytic surfaces with two independent meromorphic functions, Proc. Nom

Acnd. Sci. 38 (1952). 319-325.

9. I. DOLGACHEV: Weighted Projective Spaces, Springer Lecture Notes in Mathemarics 956 Group actions and

10

11

12

13 14

15

16

17

18

19

20

21

22.

23. 24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37

38.

39.

40.

oecrorfields, Springer, New Y&k. 1982; pp. 34-71.

I. DOLGACHEV: On the link space ofa Gorenstein quasihomogeneous surface singularity, Math. Ann. 265 (1983). 529-540. S. K. DONALDSON: An application of gauge theory to 4dimensional topology, J. Ofl. Geom. 18 (1983), 279-316.

F. ENR~Q~XS: Sulla classificazione delle superficie algebriche e particolarmente sulle superficie die genere p’ = 1,

Atci. Accad. Naz. Lincei 23 (1914). R. 0. FlLlPKIfwtCZ: Four dimensional geomerries, Ph.D. thesis, University of Warwick, 1984.

M. FREEDMAN: The topology of 4dimensional manifolds, J. D$ Geom. 17 (1982). 357-454.

R. C. GUNNING: On Liniformizarion ofComplex Manifolds; the Role ofconnections, Princeton University Press.

Princeton, NJ, 1978.

R. V. GL~RJAR and A. R. SHASTRI: Covering spaces of an elliptic surface, Compositio Math. 54 (1985x 95-104. F. E. P. HIRZEBRUCH: &er eine Klasse von einfach-zusammenhingenden komplexen Mannigfaltigkeiten, Math.

Ann. 124 (1951). 77-86.

F. E. P. HIRZEBRL.CH: Hilbert modular surfaces, Enseignement Math. 19 (19731, 183281.

F. E. P. HIRZEBRUCH: Arrangements of lines and algebraic surfaces. Progr. Math. 36 (1983). 113-140.

H. HOPF: Zur Topologie der komplexen Mannigfaltigkeiten, in Studies & Essays Presenred to R. Courant, Interrience, New York. 1948, pp. 167-185.

S. IITAKA: Deformations ofcompact complex surfaces. I: In Global Analysis: Papers in Honour of K. Kodaira, D.

C. SPENCER, ed., University ofTokyo Press and Princeton University Press. 1969, pp. 267-272. II: J. Math. Sot. Jap. 22 (1970). 247-261. III: ibid. 23 (1971). 692-704.

IITAKA: On Ddimensions of algebraic varieties, J. Math. Sot. lap. 23 (1971). 356373. M. INOUE: On surfaces of class VII,,, Incent. Marh. 24 (1974), 269-310.

M. INOUE: New surfaces with no meromorphic functions. I: Proc. Inc. Congr. Math. Vancouver. 1974, 1976,

pp. 423426. II: In Complex Analysis and Algebraic Geometry, W. L. BAILY and T. SHIODA~~~, Iwanami Shoten,

1977, pp. 91-106

M. INOUE, S. KOBAYASHI and T. OCHIAI: Holomorphic affine connexions on compact complex surfaces. J. Fat. Sci. Univ. Tokyo (Mach.) 27 (1980). 247-264.

M. KATO: Topology of Hopf surfaces, J. Math. Sot. Jap. 27 (1975), 222-238. S. KOBAYASHI and K. NOMIZU: Foundations of Differential Geometry, Vols. 1 and 2, Interscience, New York, 1963, 1969.

K. KODAIRA: On compact complex analytic surfaces. I: Ann. Math. 71(1960), I1 l-152. II: ibid. 77 (1963), 563-626. III: ibid. 78 (1963), l-40.

K. KODAIRA: On the structure of compact complex analytic surfaces. I: Am. J. Mach. 86 (1964), 751-798. II: ibid. 88 (1966). 682-721. III: ibid. 90 (1968). 55-83. IV: ibid. 90 (1968). 1048-1066. H. MAEHARA: On elliptic surfaces whose first Betti numbers are odd, in International Symposium on Algebraic Geometry. Kyoco, 1977, Kinokuniya, 1978, pp. 565-574.

R. MANDELBAUM: Surfaces of general type with positive signature, in Singularities, P. ORLIK, ed., Proc. Symp. in Pure Math., Vol. 40, Parr 2, Am. Math. Sot., 1983, pp. 193-197. Y. MIYAOKA: KPhler metrics on elliptic surfaces, Proc. Jap. Acad. SO (1974). 533-536. Y. MIYAOKA: On the Chern numbers of surfaces of genera1 type, Invent. Math. 42 (1977). 225-237.

D. MUMFORD: Geometric fnuarianr Theory, Springer, New York, 1965.

I. NAKAMURA: On surfaces of class VII,, with curves, Proc. Jap. Acad. %A (1982), 380-383; fncent. Marh. 78

(1984), 393-444.

M. NARASIMHAN and C. SESHADRI: Stable and unitary vector bundles over a compact Riemann surface, Ann. Math. 82 (1965). 540-567.

W. D. NELMA~‘N and F. RAYMOND: Seifert Manifolds, Plumbing, p-Invariant and Orientation-reversing Maps,

Springer Lecrrtre Notes in ‘Vfarhemarics 664, Algebraic and geometric topology, Springer, New York, 1978, pp. 162-196. W. D. NEUMANN: Geometry of quasihomogeneous surface singularities, in Singularities, P. ORLIK, ed., hoc. Symp. in Pure Math., Vol. 40, Parr 2, Amer. Math. Sot.. 1983, pp. 245-258.

P. ORLIK and P. WAGREICH: Isolated singularities of algebraic surfaces with I)*-action. Ann. Math. 93 (1971), 205228.

M. S. RAGHUNATHAN: Discrete Subgroups of Lie Groups, Springer, New York, 1972.

GEOMETRIC STRUCTURES ON COMPACT COMPLEX ANALYTIC SURFACES I53

41. I. SATAKE: On a generalization of the notion of manifold. Proc. Nom. Acud. Sci. 42 (1956). 35%363.

42. G. P. SCOT: The geometries of 3-manifolds. Bull. London Motfi. Sot. 15 (1983). 401387.

43. T. SHIODA: On elliptic modular surfaces, J. .Warlr. Sot. Jap. 24 (1972). 2&59.

44. Y.-T. SIU: Every 1(3 surface is Kihler, Incenr. Math. 73 (1983), 139-150.

45. T. St_rwa: Ruled surfaces of genus 1, J. Math. Sot. Jnp. 21 (1969), 291-311.

46. T. Suw~: Compact quotient spaces of C’ by affine transformation groups, J. 01% Geom. 10 (1975). 23%252. 47. W. THURSTON: 7he geometry and topolog): of 3-mnnijolds, Notes, Princeton University, 1979.

48. K. UENO: Classification ?7zheory o/ Algebric Varieties and Compact Complex Spacer (with appendix on

classification of surfaces), Springer Lecture Notes in Mathematics 439, Springer, New York, 1975.

49. C. T. C. WALL: Geometries and geometric structures in real dimension 4 and complex dimension 2, To appear in

Proceedings of University of Maryland Special Year in Low-Dimensional Topology. 50. S.-T. YAU: Calabi’s conjecture and some new results in algebraic geometry, Proc. Nom Acad. Sci. 74 (1977).

179%1799.

51. S. ZUCKER and D. A. Cox: Intersection numbers of sections of elliptic surfaces, Invent. Math. 53 (1979). l-44.

University of Liverpool

P.O. Box 147

Liverpool

L69 3BX, U.K.